@misc{hoffman_robust_2019,
	title = {Robust {Learning} with {Jacobian} {Regularization}},
	url = {http://arxiv.org/abs/1908.02729},
	abstract = {Design of reliable systems must guarantee stability against input perturbations. In machine learning, such guarantee entails preventing overﬁtting and ensuring robustness of models against corruption of input data. In order to maximize stability, we analyze and develop a computationally efﬁcient implementation of Jacobian regularization that increases classiﬁcation margins of neural networks. The stabilizing effect of the Jacobian regularizer leads to signiﬁcant improvements in robustness, as measured against both random and adversarial input perturbations, without severely degrading generalization properties on clean data.},
	language = {en},
	urldate = {2024-05-30},
	publisher = {arXiv},
	author = {Hoffman, Judy and Roberts, Daniel A. and Yaida, Sho},
	month = aug,
	year = {2019},
	note = {arXiv:1908.02729 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{tancik_fourier_2020,
	title = {Fourier {Features} {Let} {Networks} {Learn} {High} {Frequency} {Functions} in {Low} {Dimensional} {Domains}},
	volume = {33},
	abstract = {We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP has impractically slow convergence to high frequency signal components. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.},
	urldate = {2024-01-30},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Tancik, Matthew and Srinivasan, Pratul and Mildenhall, Ben and Fridovich-Keil, Sara and Raghavan, Nithin and Singhal, Utkarsh and Ramamoorthi, Ravi and Barron, Jonathan and Ng, Ren},
	year = {2020},
	pages = {7537--7547},
}

@article{yu_climsim_2023,
	title = {{ClimSim}: {A} large multi-scale dataset for hybrid physics-{ML} climate emulation},
	volume = {36},
	shorttitle = {{ClimSim}},
	language = {en},
	urldate = {2024-05-26},
	journal = {Advances in Neural Information Processing Systems},
	author = {Yu, Sungduk and Hannah, Walter and Peng, Liran and Lin, Jerry and Bhouri, Mohamed Aziz and Gupta, Ritwik and Lütjens, Björn and Will, Justus C. and Behrens, Gunnar and Busecke, Julius and Loose, Nora and Stern, Charles and Beucler, Tom and Harrop, Bryce and Hillman, Benjamin and Jenney, Andrea and Ferretti, Savannah L. and Liu, Nana and Anandkumar, Animashree and Brenowitz, Noah and Eyring, Veronika and Geneva, Nicholas and Gentine, Pierre and Mandt, Stephan and Pathak, Jaideep and Subramaniam, Akshay and Vondrick, Carl and Yu, Rose and Zanna, Laure and Zheng, Tian and Abernathey, Ryan and Ahmed, Fiaz and Bader, David and Baldi, Pierre and Barnes, Elizabeth and Bretherton, Christopher and Caldwell, Peter and Chuang, Wayne and Han, Yilun and Huang, Yu and Iglesias-Suarez, Fernando and Jantre, Sanket and Kashinath, Karthik and Khairoutdinov, Marat and Kurth, Thorsten and Lutsko, Nicholas and Ma, Po-Lun and Mooers, Griffin and Neelin, J. David and Randall, David and Shamekh, Sara and Taylor, Mark and Urban, Nathan and Yuval, Janni and Zhang, Guang and Pritchard, Mike},
	month = dec,
	year = {2023},
	pages = {22070--22084},
}

@inproceedings{bamler_improving_2018,
	title = {Improving optimization for models with continuous symmetry breaking},
	booktitle = {International {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Bamler, Robert and Mandt, Stephan},
	year = {2018},
	pages = {423--432},
}

@inproceedings{papamarkou_position_2024,
	title = {Position {Paper}: {Bayesian} {Deep} {Learning} in the {Age} of {Large}-{Scale} {AI}},
	booktitle = {International {Conference} on {Machine} {Learning}},
	author = {Papamarkou, Theodore and Skoularidou, Maria and Palla, Konstantina and Aitchison, Laurence and Arbel, Julyan and Dunson, David and Filippone, Maurizio and Fortuin, Vincent and Hennig, Philipp and Hubin, Aliaksandr and {others}},
	year = {2024},
}

@inproceedings{mandt_variational_2016,
	title = {Variational tempering},
	booktitle = {Artificial intelligence and statistics},
	publisher = {PMLR},
	author = {Mandt, Stephan and McInerney, James and Abrol, Farhan and Ranganath, Rajesh and Blei, David},
	year = {2016},
	pages = {704--712},
}

@inproceedings{stankeviciute_conformal_2021,
	title = {Conformal {Time}-series {Forecasting}},
	volume = {34},
	url = {https://proceedings.neurips.cc/paper_files/paper/2021/file/312f1ba2a72318edaaa995a67835fad5-Paper.pdf},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Stankeviciute, Kamile and M. Alaa, Ahmed and van der Schaar, Mihaela},
	editor = {Ranzato, M. and Beygelzimer, A. and Dauphin, Y. and Liang, P. S. and Vaughan, J. Wortman},
	year = {2021},
	pages = {6216--6228},
}

@article{dunn_multiple_1961,
	title = {Multiple {Comparisons} among {Means}},
	volume = {56},
	issn = {0162-1459, 1537-274X},
	url = {http://www.tandfonline.com/doi/abs/10.1080/01621459.1961.10482090},
	doi = {10.1080/01621459.1961.10482090},
	language = {en},
	number = {293},
	urldate = {2024-05-21},
	journal = {Journal of the American Statistical Association},
	author = {Dunn, Olive Jean},
	month = mar,
	year = {1961},
	pages = {52--64},
}

@article{lei_distribution-free_2018,
	title = {Distribution-{Free} {Predictive} {Inference} for {Regression}},
	volume = {113},
	issn = {0162-1459},
	url = {https://doi.org/10.1080/01621459.2017.1307116},
	doi = {10.1080/01621459.2017.1307116},
	abstract = {We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We analyze and compare, both empirically and theoretically, the two major variants of our conformal framework: full conformal inference and split conformal inference, along with a related jackknife method. These methods offer different tradeoffs between statistical accuracy (length of resulting prediction intervals) and computational efficiency. As extensions, we develop a method for constructing valid in-sample prediction intervals called rank-one-out conformal inference, which has essentially the same computational efficiency as split conformal inference. We also describe an extension of our procedures for producing prediction bands with locally varying length, to adapt to heteroscedasticity in the data. Finally, we propose a model-free notion of variable importance, called leave-one-covariate-out or LOCO inference. Accompanying this article is an R package conformalInference that implements all of the proposals we have introduced. In the spirit of reproducibility, all of our empirical results can also be easily (re)generated using this package.},
	number = {523},
	urldate = {2024-05-20},
	journal = {Journal of the American Statistical Association},
	author = {Lei, Jing and G’Sell, Max and Rinaldo, Alessandro and Tibshirani, Ryan J. and Wasserman, Larry},
	month = jul,
	year = {2018},
	note = {Publisher: Taylor \& Francis
\_eprint: https://doi.org/10.1080/01621459.2017.1307116},
	keywords = {Distribution-free, Model misspecification, Prediction band, Regression, Variable importance},
	pages = {1094--1111},
}

@inproceedings{tancik_fourier_2020-1,
	title = {Fourier {Features} {Let} {Networks} {Learn} {High} {Frequency} {Functions} in {Low} {Dimensional} {Domains}},
	volume = {33},
	url = {https://proceedings.neurips.cc/paper/2020/hash/55053683268957697aa39fba6f231c68-Abstract.html},
	abstract = {We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP has impractically slow convergence to high frequency signal components. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.},
	urldate = {2024-01-30},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Tancik, Matthew and Srinivasan, Pratul and Mildenhall, Ben and Fridovich-Keil, Sara and Raghavan, Nithin and Singhal, Utkarsh and Ramamoorthi, Ravi and Barron, Jonathan and Ng, Ren},
	year = {2020},
	pages = {7537--7547},
}

@article{bishop_training_1995,
	title = {Training with {Noise} is {Equivalent} to {Tikhonov} {Regularization}},
	volume = {7},
	issn = {0899-7667, 1530-888X},
	url = {https://direct.mit.edu/neco/article/7/1/108-116/5828},
	doi = {10.1162/neco.1995.7.1.108},
	abstract = {It is well known that the addition of noise to the input data of a neural network during training can, in some circumstances, lead to signiﬁcant improvements in generalization performance. Previous work has shown that such training with noise is equivalent to a form of regularization in which an extra term is added to the error function. However, the regularization term, which involves second derivatives of the error function, is not bounded below, and so can lead to diﬃculties if used directly in a learning algorithm based on error minimization. In this paper we show that, for the purposes of network training, the regularization term can be reduced to a positive deﬁnite form which involves only ﬁrst derivatives of the network mapping. For a sum-of-squares error function, the regularization term belongs to the class of generalized Tikhonov regularizers. Direct minimization of the regularized error function provides a practical alternative to training with noise.},
	language = {en},
	number = {1},
	urldate = {2023-10-24},
	journal = {Neural Computation},
	author = {Bishop, Chris M.},
	month = jan,
	year = {1995},
	pages = {108--116},
}

@article{hanin_how_nodate,
	title = {How {Data} {Augmentation} affects {Optimization} for {Linear} {Regression}},
	abstract = {Though data augmentation has rapidly emerged as a key tool for optimization in modern machine learning, a clear picture of how augmentation schedules affect optimization and interact with optimization hyperparameters such as learning rate is nascent. In the spirit of classical convex optimization and recent work on implicit bias, the present work analyzes the effect of augmentation on optimization in the simple convex setting of linear regression with MSE loss.},
	language = {en},
	author = {Hanin, Boris and Sun, Yi},
}

@misc{venkatesh_heteroscedastic_2019,
	title = {Heteroscedastic {Calibration} of {Uncertainty} {Estimators} in {Deep} {Learning}},
	url = {http://arxiv.org/abs/1910.14179},
	abstract = {The role of uncertainty quantification (UQ) in deep learning has become crucial with growing use of predictive models in high-risk applications. Though a large class of methods exists for measuring deep uncertainties, in practice, the resulting estimates are found to be poorly calibrated, thus making it challenging to translate them into actionable insights. A common workaround is to utilize a separate recalibration step, which adjusts the estimates to compensate for the miscalibration. Instead, we propose to repurpose the heteroscedastic regression objective as a surrogate for calibration and enable any existing uncertainty estimator to be inherently calibrated. In addition to eliminating the need for recalibration, this also regularizes the training process. Using regression experiments, we demonstrate the effectiveness of the proposed heteroscedastic calibration with two popular uncertainty estimators.},
	urldate = {2023-10-24},
	publisher = {arXiv},
	author = {Venkatesh, Bindya and Thiagarajan, Jayaraman J.},
	month = oct,
	year = {2019},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{hastie_ridge_2020,
	title = {Ridge {Regularization}: {An} {Essential} {Concept} in {Data} {Science}},
	volume = {62},
	issn = {0040-1706},
	shorttitle = {Ridge {Regularization}},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9410599/},
	doi = {10.1080/00401706.2020.1791959},
	abstract = {Ridge or more formally ℓ2 regularization shows up in many areas of statistics and machine learning. It is one of those essential devices that any good data scientist needs to master for their craft. In this brief ridge fest, I have collected together some of the magic and beauty of ridge that my colleagues and I have encountered over the past 40 years in applied statistics.},
	number = {4},
	urldate = {2023-10-13},
	journal = {Technometrics : a journal of statistics for the physical, chemical, and engineering sciences},
	author = {Hastie, Trevor},
	year = {2020},
	pmid = {36033922},
	pmcid = {PMC9410599},
	pages = {426--433},
}

@misc{sluijterman_optimal_2023,
	title = {Optimal {Training} of {Mean} {Variance} {Estimation} {Neural} {Networks}},
	url = {http://arxiv.org/abs/2302.08875},
	abstract = {This paper focusses on the optimal implementation of a Mean Variance Estimation network (MVE network) (Nix and Weigend, 1994). This type of network is often used as a building block for uncertainty estimation methods in a regression setting, for instance Concrete dropout (Gal et al., 2017) and Deep Ensembles (Lakshminarayanan et al., 2017). Specifically, an MVE network assumes that the data is produced from a normal distribution with a mean function and variance function. The MVE network outputs a mean and variance estimate and optimizes the network parameters by minimizing the negative loglikelihood. In our paper, we present two significant insights. Firstly, the convergence difficulties reported in recent work can be relatively easily prevented by following the simple yet often overlooked recommendation from the original authors that a warm-up period should be used. During this period, only the mean is optimized with a fixed variance. We demonstrate the effectiveness of this step through experimentation, highlighting that it should be standard practice. As a sidenote, we examine whether, after the warm-up, it is beneficial to fix the mean while optimizing the variance or to optimize both simultaneously. Here, we do not observe a substantial difference. Secondly, we introduce a novel improvement of the MVE network: separate regularization of the mean and the variance estimate. We demonstrate, both on toy examples and on a number of benchmark UCI regression data sets, that following the original recommendations and the novel separate regularization can lead to significant improvements.},
	urldate = {2023-10-04},
	publisher = {arXiv},
	author = {Sluijterman, Laurens and Cator, Eric and Heskes, Tom},
	month = aug,
	year = {2023},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{yu_climsim_2023-1,
	title = {{ClimSim}: {An} open large-scale dataset for training high-resolution physics emulators in hybrid multi-scale climate simulators},
	shorttitle = {{ClimSim}},
	url = {http://arxiv.org/abs/2306.08754},
	abstract = {Modern climate projections lack adequate spatial and temporal resolution due to computational constraints. A consequence is inaccurate and imprecise predictions of critical processes such as storms. Hybrid methods that combine physics with machine learning (ML) have introduced a new generation of higher fidelity climate simulators that can sidestep Moore's Law by outsourcing compute-hungry, short, high-resolution simulations to ML emulators. However, this hybrid ML-physics simulation approach requires domain-specific treatment and has been inaccessible to ML experts because of lack of training data and relevant, easy-to-use workflows. We present ClimSim, the largest-ever dataset designed for hybrid ML-physics research. It comprises multi-scale climate simulations, developed by a consortium of climate scientists and ML researchers. It consists of 5.7 billion pairs of multivariate input and output vectors that isolate the influence of locally-nested, high-resolution, high-fidelity physics on a host climate simulator's macro-scale physical state. The dataset is global in coverage, spans multiple years at high sampling frequency, and is designed such that resulting emulators are compatible with downstream coupling into operational climate simulators. We implement a range of deterministic and stochastic regression baselines to highlight the ML challenges and their scoring. The data (https://huggingface.co/datasets/LEAP/ClimSim\_high-res, https://huggingface.co/datasets/LEAP/ClimSim\_low-res, and https://huggingface.co/datasets/LEAP/ClimSim\_low-res\_aqua-planet) and code (https://leap-stc.github.io/ClimSim) are released openly to support the development of hybrid ML-physics and high-fidelity climate simulations for the benefit of science and society.},
	urldate = {2023-09-28},
	publisher = {arXiv},
	author = {Yu, Sungduk and Hannah, Walter M. and Peng, Liran and Lin, Jerry and Bhouri, Mohamed Aziz and Gupta, Ritwik and Lütjens, Björn and Will, Justus C. and Behrens, Gunnar and Busecke, Julius J. M. and Loose, Nora and Stern, Charles and Beucler, Tom and Harrop, Bryce E. and Hilman, Benjamin R. and Jenney, Andrea M. and Ferretti, Savannah L. and Liu, Nana and Anandkumar, Anima and Brenowitz, Noah D. and Eyring, Veronika and Geneva, Nicholas and Gentine, Pierre and Mandt, Stephan and Pathak, Jaideep and Subramaniam, Akshay and Vondrick, Carl and Yu, Rose and Zanna, Laure and Zheng, Tian and Abernathey, Ryan P. and Ahmed, Fiaz and Bader, David C. and Baldi, Pierre and Barnes, Elizabeth A. and Bretherton, Christopher S. and Caldwell, Peter M. and Chuang, Wayne and Han, Yilun and Huang, Yu and Iglesias-Suarez, Fernando and Jantre, Sanket and Kashinath, Karthik and Khairoutdinov, Marat and Kurth, Thorsten and Lutsko, Nicholas J. and Ma, Po-Lun and Mooers, Griffin and Neelin, J. David and Randall, David A. and Shamekh, Sara and Taylor, Mark A. and Urban, Nathan M. and Yuval, Janni and Zhang, Guang J. and Pritchard, Michael S.},
	month = sep,
	year = {2023},
	doi = {10.48550/arXiv.2306.08754},
	keywords = {Computer Science - Machine Learning, Physics - Atmospheric and Oceanic Physics},
}

@inproceedings{kuleshov_accurate_2018,
	title = {Accurate {Uncertainties} for {Deep} {Learning} {Using} {Calibrated} {Regression}},
	abstract = {Methods for reasoning under uncertainty are a key building block of accurate and reliable machine learning systems. Bayesian methods provide a general framework to quantify uncertainty. However, because of model misspeciﬁcation and the use of approximate inference, Bayesian uncertainty estimates are often inaccurate — for example, a 90\% credible interval may not contain the true outcome 90\% of the time. Here, we propose a simple procedure for calibrating any regression algorithm; when applied to Bayesian and probabilistic models, it is guaranteed to produce calibrated uncertainty estimates given enough data. Our procedure is inspired by Platt scaling and extends previous work on classiﬁcation. We evaluate this approach on Bayesian linear regression, feedforward, and recurrent neural networks, and ﬁnd that it consistently outputs well-calibrated credible intervals while improving performance on time series forecasting and model-based reinforcement learning tasks.},
	language = {en},
	author = {Kuleshov, Volodymyr and Fenner, Nathan and Ermon, Stefano},
	year = {2018},
}

@inproceedings{chung_beyond_2021,
	title = {Beyond {Pinball} {Loss}: {Quantile} {Methods} for {Calibrated} {Uncertainty} {Quantiﬁcation}},
	abstract = {Among the many ways of quantifying uncertainty in a regression setting, specifying the full quantile function is attractive, as quantiles are amenable to interpretation and evaluation. A model that predicts the true conditional quantiles for each input, at all quantile levels, presents a correct and efﬁcient representation of the underlying uncertainty. To achieve this, many current quantile-based methods focus on optimizing the pinball loss. However, this loss restricts the scope of applicable regression models, limits the ability to target many desirable properties (e.g. calibration, sharpness, centered intervals), and may produce poor conditional quantiles. In this work, we develop new quantile methods that address these shortcomings. In particular, we propose methods that can apply to any class of regression model, select an explicit balance between calibration and sharpness, optimize for calibration of centered intervals, and produce more accurate conditional quantiles. We provide a thorough experimental evaluation of our methods, which includes a high dimensional uncertainty quantiﬁcation task in nuclear fusion.},
	language = {en},
	author = {Chung, Youngseog and Neiswanger, Willie},
	year = {2021},
}

@misc{noauthor_parallel_nodate,
	title = {parallel - {Jupyter} {Notebook}},
	url = {http://localhost:6006/notebooks/parallel.ipynb},
	urldate = {2023-09-28},
}

@article{levi_evaluating_2022,
	title = {Evaluating and {Calibrating} {Uncertainty} {Prediction} in {Regression} {Tasks}},
	volume = {22},
	issn = {1424-8220},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9330317/},
	doi = {10.3390/s22155540},
	abstract = {Predicting not only the target but also an accurate measure of uncertainty is important for many machine learning applications, and in particular, safety-critical ones. In this work, we study the calibration of uncertainty prediction for regression tasks which often arise in real-world systems. We show that the existing definition for the calibration of regression uncertainty has severe limitations in distinguishing informative from non-informative uncertainty predictions. We propose a new definition that escapes this caveat and an evaluation method using a simple histogram-based approach. Our method clusters examples with similar uncertainty prediction and compares the prediction with the empirical uncertainty on these examples. We also propose a simple, scaling-based calibration method that preforms as well as much more complex ones. We show results on both a synthetic, controlled problem and on the object detection bounding-box regression task using the COCO and KITTI datasets.},
	number = {15},
	urldate = {2023-09-27},
	journal = {Sensors (Basel, Switzerland)},
	author = {Levi, Dan and Gispan, Liran and Giladi, Niv and Fetaya, Ethan},
	month = jul,
	year = {2022},
	pmid = {35898047},
	pmcid = {PMC9330317},
	pages = {5540},
}

@misc{kelly_uci_nodate,
	title = {The {UCI} {Machine} {Learning} {Repository}},
	url = {https://archive.ics.uci.edu},
	publisher = {The UCI Machine Learning Repository},
	author = {Kelly, Markelle and Longjohn, Rachel and Nottingham, Kolby},
}

@misc{yeh_concrete_2007,
	title = {Concrete {Compressive} {Strength}},
	publisher = {UCI Machine Learning Repository},
	author = {Yeh, I-Cheng},
	year = {2007},
	doi = {10.24432/C5PK67},
}

@misc{noauthor_uci_nodate,
	title = {{UCI} {Machine} {Learning} {Repository}},
	urldate = {2023-09-27},
}

@article{harrison_hedonic_1978,
	title = {Hedonic housing prices and the demand for clean air},
	volume = {5},
	issn = {0095-0696},
	doi = {10.1016/0095-0696(78)90006-2},
	abstract = {This paper investigates the methodological problems associated with the use of housing market data to measure the willingness to pay for clean air. With the use of a hedonic housing price model and data for the Boston metropolitan area, quantitative estimates of the willingness to pay for air quality improvements are generated. Marginal air pollution damages (as revealed in the housing market) are found to increase with the level of air pollution and with household income. The results are relatively sensitive to the specification of the hedonic housing price equation, but insensitive to the specification of the air quality demand equation.},
	number = {1},
	urldate = {2023-09-27},
	journal = {Journal of Environmental Economics and Management},
	author = {Harrison, David and Rubinfeld, Daniel L},
	month = mar,
	year = {1978},
	pages = {81--102},
}

@inproceedings{stirn_faithful_2023,
	title = {Faithful {Heteroscedastic} {Regression} with {Neural} {Networks}},
	abstract = {Heteroscedastic regression models a Gaussian variable’s mean and variance as a function of covariates. Parametric methods that employ neural networks for these parameter maps can capture complex relationships in the data. Yet, optimizing network parameters via log likelihood gradients can yield suboptimal mean and uncalibrated variance estimates. Current solutions side-step this optimization problem with surrogate objectives or Bayesian treatments. Instead, we make two simple modiﬁcations to optimization. Notably, their combination produces a heteroscedastic model with mean estimates that are provably as accurate as those from its homoscedastic counterpart (i.e. ﬁtting the mean under squared error loss). For a wide variety of network and task complexities, we ﬁnd that mean estimates from existing heteroscedastic solutions can be signiﬁcantly less accurate than those from an equivalently expressive mean-only model. Our approach provably retains the accuracy of an equally ﬂexible mean-only model while also offering best-in-class variance calibration. Lastly, we show how to leverage our method to recover the underlying heteroscedastic noise variance.},
	language = {en},
	author = {Stirn, Andrew and Wessels, Hans-Hermann and Schertzer, Megan and Pereira, Laura and Sanjana, Neville E and Knowles, David A},
	year = {2023},
}

@inproceedings{liu_simple_2020,
	title = {Simple and {Principled} {Uncertainty} {Estimation} with {Deterministic} {Deep} {Learning} via {Distance} {Awareness}},
	language = {en},
	author = {Liu, Jeremiah Zhe and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax-Weiss, Tania and Lakshminarayanan, Balaji},
	year = {2020},
}

@inproceedings{liu_simple_2020-1,
	title = {Simple and {Principled} {Uncertainty} {Estimation} with {Deterministic} {Deep} {Learning} via {Distance} {Awareness}},
	volume = {33},
	url = {https://proceedings.neurips.cc/paper/2020/hash/543e83748234f7cbab21aa0ade66565f-Abstract.html},
	abstract = {Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model’s ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.},
	urldate = {2023-09-27},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Liu, Jeremiah and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax Weiss, Tania and Lakshminarayanan, Balaji},
	year = {2020},
	pages = {7498--7512},
}

@inproceedings{lakshminarayanan_simple_2017,
	title = {Simple and {Scalable} {Predictive} {Uncertainty} {Estimation} using {Deep} {Ensembles}},
	abstract = {Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs, which learn a distribution over weights, are currently the state-of-the-art for estimating predictive uncertainty; however these require signiﬁcant modiﬁcations to the training procedure and are computationally expensive compared to standard (non-Bayesian) NNs. We propose an alternative to Bayesian NNs that is simple to implement, readily parallelizable, requires very little hyperparameter tuning, and yields high quality predictive uncertainty estimates. Through a series of experiments on classiﬁcation and regression benchmarks, we demonstrate that our method produces well-calibrated uncertainty estimates which are as good or better than approximate Bayesian NNs. To assess robustness to dataset shift, we evaluate the predictive uncertainty on test examples from known and unknown distributions, and show that our method is able to express higher uncertainty on out-of-distribution examples. We demonstrate the scalability of our method by evaluating predictive uncertainty estimates on ImageNet.},
	language = {en},
	booktitle = {Neural {Information} {Processing} {Systems}},
	author = {Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles},
	year = {2017},
}

@inproceedings{skafte_reliable_2019,
	title = {Reliable training and estimation of variance networks},
	abstract = {We propose and investigate new complementary methodologies for estimating predictive variance networks in regression neural networks. We derive a locally aware mini-batching scheme that results in sparse robust gradients, and we show how to make unbiased weight updates to a variance network. Further, we formulate a heuristic for robustly ﬁtting both the mean and variance networks post hoc. Finally, we take inspiration from posterior Gaussian processes and propose a network architecture with similar extrapolation properties to Gaussian processes. The proposed methodologies are complementary, and improve upon baseline methods individually. Experimentally, we investigate the impact of predictive uncertainty on multiple datasets and tasks ranging from regression, active learning and generative modeling. Experiments consistently show signiﬁcant improvements in predictive uncertainty estimation over state-of-the-art methods across tasks and datasets.},
	language = {en},
	author = {Skafte, Nicki and Jørgensen, Martin and Hauberg, Søren},
	year = {2019},
}

@book{landau_statistical_2013,
	title = {Statistical {Physics}: {Volume} 5},
	isbn = {978-0-08-057046-4},
	shorttitle = {Statistical {Physics}},
	abstract = {A lucid presentation of statistical physics and thermodynamics which develops from the general principles to give a large number of applications of the theory.},
	language = {en},
	publisher = {Elsevier},
	author = {Landau, L. D. and Lifshitz, E. M.},
	month = oct,
	year = {2013},
	keywords = {Mathematics / Probability \& Statistics / General, Science / Physics / General, Science / Physics / Mathematical \& Computational},
}

@article{fortuin_deep_2022,
	title = {Deep {Classiﬁers} with {Label} {Noise} {Modeling} and {Distance} {Awareness}},
	abstract = {Uncertainty estimation in deep learning has recently emerged as a crucial area of interest to advance reliability and robustness in safety-critical applications. While there have been many proposed methods that either focus on distance-aware model uncertainties for out-of-distribution detection or on input-dependent label uncertainties for in-distribution calibration, both of these types of uncertainty are often necessary. In this work, we propose the HetSNGP method for jointly modeling the model and data uncertainty. We show that our proposed model aﬀords a favorable combination between these two types of uncertainty and thus outperforms the baseline methods on some challenging out-of-distribution datasets, including CIFAR-100C, ImageNet-C, and ImageNet-A. Moreover, we propose HetSNGP Ensemble, an ensembled version of our method which additionally models uncertainty over the network parameters and outperforms other ensemble baselines.},
	language = {en},
	journal = {Transactions on Machine Learning Research},
	author = {Fortuin, Vincent and Collier, Mark and Wenzel, Florian and Allingham, James and Liu, Jeremiah and Tran, Dustin and Lakshminarayanan, Balaji and Berent, Jesse and Jenatton, Rodolphe and Kokiopoulou, Eﬀrosyni},
	year = {2022},
}

@inproceedings{wang_robust_2017,
	address = {Sydney, NSW, Australia},
	series = {{ICML}'17},
	title = {Robust probabilistic modeling with {Bayesian} data reweighting},
	abstract = {Probabilistic models analyze data by relying on a set of assumptions. Data that exhibit deviations from these assumptions can undermine inference and prediction quality. Robust models offer protection against mismatch between a model's assumptions and reality. We propose a way to systematically detect and mitigate mismatch of a large class of probabilistic models. The idea is to raise the likelihood of each observation to a weight and then to infer both the latent variables and the weights from data. Inferring the weights allows a model to identify observations that match its assumptions and down-weight others. This enables robust inference and improves predictive accuracy. We study four different forms of mismatch with reality, ranging from missing latent groups to structure misspecification. A Poisson factorization analysis of the Movielens 1M dataset shows the benefits of this approach in a practical scenario.},
	urldate = {2023-09-27},
	booktitle = {Proceedings of the 34th {International} {Conference} on {Machine} {Learning} - {Volume} 70},
	publisher = {JMLR.org},
	author = {Wang, Yixin and Kucukelbir, Alp and Blei, David M.},
	month = aug,
	year = {2017},
	pages = {3646--3655},
}

@inproceedings{collier_correlated_2021,
	address = {Nashville, TN, USA},
	title = {Correlated {Input}-{Dependent} {Label} {Noise} in {Large}-{Scale} {Image} {Classification}},
	isbn = {978-1-66544-509-2},
	doi = {10.1109/CVPR46437.2021.00160},
	abstract = {Large scale image classiﬁcation datasets often contain noisy labels. We take a principled probabilistic approach to modelling input-dependent, also known as heteroscedastic, label noise in these datasets. We place a multivariate Normal distributed latent variable on the ﬁnal hidden layer of a neural network classiﬁer. The covariance matrix of this latent variable, models the aleatoric uncertainty due to label noise. We demonstrate that the learned covariance structure captures known sources of label noise between semantically similar and co-occurring classes. Compared to standard neural network training and other baselines, we show significantly improved accuracy on Imagenet ILSVRC 2012 79.3\% (+ 2.6\%), Imagenet-21k 47.0\% (+ 1.1\%) and JFT 64.7\% (+ 1.6\%). We set a new state-of-the-art result on WebVision 1.0 with 76.6\% top-1 accuracy. These datasets range from over 1M to over 300M training examples and from 1k classes to more than 21k classes. Our method is simple to use, and we provide an implementation that is a drop-in replacement for the ﬁnal fully-connected layer in a deep classiﬁer.},
	language = {en},
	urldate = {2023-06-20},
	booktitle = {2021 {IEEE}/{CVF} {Conference} on {Computer} {Vision} and {Pattern} {Recognition} ({CVPR})},
	publisher = {IEEE},
	author = {Collier, Mark and Mustafa, Basil and Kokiopoulou, Efi and Jenatton, Rodolphe and Berent, Jesse},
	month = jun,
	year = {2021},
	pages = {1551--1560},
}

@article{seitzer_pitfalls_2022,
	title = {{ON} {THE} {PITFALLS} {OF} {HETEROSCEDASTIC} {UNCERTAINTY} {ESTIMATION} {WITH} {PROBABILISTIC} {NEURAL} {NETWORKS}},
	abstract = {Capturing aleatoric uncertainty is a critical part of many machine learning systems. In deep learning, a common approach to this end is to train a neural network to estimate the parameters of a heteroscedastic Gaussian distribution by maximizing the logarithm of the likelihood function under the observed data. In this work, we examine this approach and identify potential hazards associated with the use of log-likelihood in conjunction with gradient-based optimizers. First, we present a synthetic example illustrating how this approach can lead to very poor but stable parameter estimates. Second, we identify the culprit to be the log-likelihood loss, along with certain conditions that exacerbate the issue. Third, we present an alternative formulation, termed β−NLL, in which each data point’s contribution to the loss is weighted by the β-exponentiated variance estimate. We show that using an appropriate β largely mitigates the issue in our illustrative example. Fourth, we evaluate this approach on a range of domains and tasks and show that it achieves considerable improvements and performs more robustly concerning hyperparameters, both in predictive RMSE and log-likelihood criteria.},
	language = {en},
	author = {Seitzer, Maximilian and Tavakoli, Arash and Antic, Dimitrije and Martius, Georg},
	year = {2022},
}

@misc{schaeffer_are_2023,
	title = {Are {Emergent} {Abilities} of {Large} {Language} {Models} a {Mirage}?},
	abstract = {Recent work claims that large language models display emergent abilities, abilities not present in smaller-scale models that are present in larger-scale models. What makes emergent abilities intriguing is two-fold: their sharpness, transitioning seemingly instantaneously from not present to present, and their unpredictability, appearing at seemingly unforeseeable model scales. Here, we present an alternative explanation for emergent abilities: that for a particular task and model family, when analyzing fixed model outputs, emergent abilities appear due to the researcher's choice of metric rather than due to fundamental changes in model behavior with scale. Specifically, nonlinear or discontinuous metrics produce apparent emergent abilities, whereas linear or continuous metrics produce smooth, continuous predictable changes in model performance. We present our alternative explanation in a simple mathematical model, then test it in three complementary ways: we (1) make, test and confirm three predictions on the effect of metric choice using the InstructGPT/GPT-3 family on tasks with claimed emergent abilities; (2) make, test and confirm two predictions about metric choices in a meta-analysis of emergent abilities on BIG-Bench; and (3) show to choose metrics to produce never-before-seen seemingly emergent abilities in multiple vision tasks across diverse deep networks. Via all three analyses, we provide evidence that alleged emergent abilities evaporate with different metrics or with better statistics, and may not be a fundamental property of scaling AI models.},
	urldate = {2023-05-25},
	publisher = {arXiv},
	author = {Schaeffer, Rylan and Miranda, Brando and Koyejo, Sanmi},
	month = may,
	year = {2023},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning},
}

@article{tufekci_prediction_2014,
	title = {Prediction of full load electrical power output of a base load operated combined cycle power plant using machine learning methods},
	volume = {60},
	issn = {0142-0615},
	doi = {10.1016/j.ijepes.2014.02.027},
	abstract = {Predicting full load electrical power output of a base load power plant is important in order to maximize the profit from the available megawatt hours. This paper examines and compares some machine learning regression methods to develop a predictive model, which can predict hourly full load electrical power output of a combined cycle power plant. The base load operation of a power plant is influenced by four main parameters, which are used as input variables in the dataset, such as ambient temperature, atmospheric pressure, relative humidity, and exhaust steam pressure. These parameters affect electrical power output, which is considered as the target variable. The dataset, which consists of these input and target variables, was collected over a six-year period. First, based on these variables the best subset of the dataset is explored among all feature subsets in the experiments. Then, the most successful machine learning regression method is sought for predicting full load electrical power output. Thus, the best performance of the best subset, which contains a complete set of input variables, has been observed using the most successful method, which is Bagging algorithm with REPTree, with a mean absolute error of 2.818 and a Root Mean-Squared Error of 3.787.},
	language = {en},
	urldate = {2023-05-17},
	journal = {International Journal of Electrical Power \& Energy Systems},
	author = {Tüfekci, Pınar},
	month = sep,
	year = {2014},
	keywords = {Combined cycle power plants, Machine learning methods, Prediction of electrical power output},
	pages = {126--140},
}

@article{gerritsma_geometry_1981,
	title = {Geometry, resistance and stability of the {Delft} {Systematic} {Yacht} hull series},
	language = {en},
	urldate = {2023-05-17},
	journal = {TU Delft, Faculty of Marine Technology, Ship Hydromechanics Laboratory, Report No. 520-P, Published in: International Shipbuilding Progress, ISP, Delft, The Netherlands, Volume 28, No. 328, also 7th HISWA Symposium, Amsterdam, The Netherlands},
	author = {Gerritsma, J.},
	year = {1981},
}

@misc{noauthor_uci_nodate-1,
	title = {{UCI} {Machine} {Learning} {Repository}: {Yacht} {Hydrodynamics} {Data} {Set}},
	url = {https://archive.ics.uci.edu/ml/datasets/yacht+hydrodynamics},
	urldate = {2023-05-17},
}

@article{hornik_approximation_1991,
	title = {Approximation capabilities of multilayer feedforward networks},
	volume = {4},
	issn = {0893-6080},
	doi = {10.1016/0893-6080(91)90009-T},
	abstract = {We show that standard multilayer feedforward networks with as few as a single hidden layer and arbitrary bounded and nonconstant activation function are universal approximators with respect to Lp(μ) performance criteria, for arbitrary finite input environment measures μ, provided only that sufficiently many hidden units are available. If the activation function is continuous, bounded and nonconstant, then continuous mappings can be learned uniformly over compact input sets. We also give very general conditions ensuring that networks with sufficiently smooth activation functions are capable of arbitrarily accurate approximation to a function and its derivatives.},
	language = {en},
	number = {2},
	urldate = {2023-05-16},
	journal = {Neural Networks},
	author = {Hornik, Kurt},
	month = jan,
	year = {1991},
	keywords = {() approximation, Activation function, Input environment measure, Multilayer feedforward networks, Smooth approximation, Sobolev spaces, Uniform approximation, Universal approximation capabilities},
	pages = {251--257},
}

@misc{noauthor_extra_nodate,
	title = {Extra {Materials} {Archive} {Search}},
	urldate = {2023-05-16},
}

@book{engel_density_2011,
	address = {Berlin, Heidelberg},
	series = {Theoretical and {Mathematical} {Physics}},
	title = {Density {Functional} {Theory}: {An} {Advanced} {Course}},
	isbn = {978-3-642-14089-1 978-3-642-14090-7},
	shorttitle = {Density {Functional} {Theory}},
	language = {en},
	urldate = {2023-05-16},
	publisher = {Springer},
	author = {Engel, Eberhard and Dreizler, Reiner M.},
	year = {2011},
	doi = {10.1007/978-3-642-14090-7},
	keywords = {Density Functional Theory, Exchange-Correlation Energy, Kohn-Sham Equations},
}

@article{astivia_heteroskedasticity_2019,
	title = {Heteroskedasticity in {Multiple} {Regression} {Analysis}: {What} it is, {How} to {Detect} it and {How} to {Solve} it with {Applications} in {R} and {SPSS}},
	volume = {24},
	issn = {1531-7714},
	shorttitle = {Heteroskedasticity in {Multiple} {Regression} {Analysis}},
	doi = {https://doi.org/10.7275/q5xr-fr95},
	number = {1},
	journal = {Practical Assessment, Research, and Evaluation},
	author = {Astivia, Oscar and Zumbo, Bruno},
	month = nov,
	year = {2019},
}

@misc{stirn_faithful_2022,
	title = {Faithful {Heteroscedastic} {Regression} with {Neural} {Networks}},
	abstract = {Heteroscedastic regression models a Gaussian variable's mean and variance as a function of covariates. Parametric methods that employ neural networks for these parameter maps can capture complex relationships in the data. Yet, optimizing network parameters via log likelihood gradients can yield suboptimal mean and uncalibrated variance estimates. Current solutions side-step this optimization problem with surrogate objectives or Bayesian treatments. Instead, we make two simple modifications to optimization. Notably, their combination produces a heteroscedastic model with mean estimates that are provably as accurate as those from its homoscedastic counterpart (i.e.{\textbackslash}textasciitildefitting the mean under squared error loss). For a wide variety of network and task complexities, we find that mean estimates from existing heteroscedastic solutions can be significantly less accurate than those from an equivalently expressive mean-only model. Our approach provably retains the accuracy of an equally flexible mean-only model while also offering best-in-class variance calibration. Lastly, we show how to leverage our method to recover the underlying heteroscedastic noise variance.},
	language = {en},
	urldate = {2023-05-12},
	author = {Stirn, Andrew and Wessels, Hans-Hermann and Schertzer, Megan and Pereira, Laura and Sanjana, Neville E. and Knowles, David A.},
	month = dec,
	year = {2022},
	note = {Publication Title: arXiv.org},
}

@article{fornberg_generation_1988,
	title = {Generation of {Finite} {Difference} {Formulas} on {Arbitrarily} {Spaced} {Grids}},
	volume = {51},
	abstract = {Simple recursions are derived for calculating the weights in compact finite difference formulas for any order of derivative and to any order of accuracy on onedimensional grids with arbitrary spacing. Tables are included for some special cases (of equispaced grids).},
	language = {en},
	journal = {Mathematics of Computation},
	author = {Fornberg, Bengt},
	month = oct,
	year = {1988},
	pages = {699--706},
}

@misc{noauthor_uci_nodate-2,
	title = {{UCI} {Machine} {Learning} {Repository}},
	url = {https://archive.ics.uci.edu/ml/index.php},
	urldate = {2023-05-07},
}

@misc{smith_cyclical_2017,
	title = {Cyclical {Learning} {Rates} for {Training} {Neural} {Networks}},
	abstract = {It is known that the learning rate is the most important hyper-parameter to tune for training deep neural networks. This paper describes a new method for setting the learning rate, named cyclical learning rates, which practically eliminates the need to experimentally find the best values and schedule for the global learning rates. Instead of monotonically decreasing the learning rate, this method lets the learning rate cyclically vary between reasonable boundary values. Training with cyclical learning rates instead of fixed values achieves improved classification accuracy without a need to tune and often in fewer iterations. This paper also describes a simple way to estimate "reasonable bounds" – linearly increasing the learning rate of the network for a few epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10 and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets, and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These are practical tools for everyone who trains neural networks.},
	urldate = {2023-03-01},
	publisher = {arXiv},
	author = {Smith, Leslie N.},
	month = apr,
	year = {2017},
	keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing},
}

@inproceedings{krizhevsky_imagenet_2012,
	title = {{ImageNet} {Classification} with {Deep} {Convolutional} {Neural} {Networks}},
	volume = {25},
	url = {https://papers.nips.cc/paper/2012/hash/c399862d3b9d6b76c8436e924a68c45b-Abstract.html},
	abstract = {We trained a large, deep convolutional neural network to classify the 1.3 million high-resolution images in the LSVRC-2010 ImageNet training set into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 39.7{\textbackslash}textbackslash\% and 18.9{\textbackslash}textbackslash\% which is considerably better than the previous state-of-the-art results. The neural network, which has 60 million parameters and 500,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and two globally connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of convolutional nets. To reduce overfitting in the globally connected layers we employed a new regularization method that proved to be very effective.},
	urldate = {2023-02-17},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Krizhevsky, Alex and Sutskever, Ilya and Hinton, Geoffrey E},
	year = {2012},
}

@article{olvera_astivia_heteroskedasticity_nodate,
	title = {Heteroskedasticity in {Multiple} {Regression} {Analysis}: {What} it is, {How} to {Detect} it and {How} to {Solve} it with {Applications} in {R} and {SPSS}},
	shorttitle = {Heteroskedasticity in {Multiple} {Regression} {Analysis}},
	url = {https://scholarworks.umass.edu/pare/vol24/iss1/1/},
	doi = {10.7275/Q5XR-FR95},
	language = {en},
	urldate = {2023-02-17},
	author = {Olvera Astivia, Oscar L and Zumbo, Bruno D},
}

@incollection{ahmad_deep_2019,
	address = {Singapore},
	series = {{SpringerBriefs} in {Computer} {Science}},
	title = {Deep {Learning} {Methods} and {Applications}},
	isbn = {9789811334597},
	url = {https://doi.org/10.1007/978-981-13-3459-7_3},
	abstract = {This chapter introduces the various methods existing beneath the umbrella of deep learning paradigm, their intricate details, and their applications in various fields. Deep learning has substantially improved the predictive capacity of computing devices, due to the availability of big data, with the help of superior learning algorithms. It has made it possible as well as practical to integrate machine learning with sophisticated applications including image recognition, object detection, self-driving cars, drug discovery, and disease detection. The superior and reliable performance of deep learning methods has attracted the attention of researchers working in every field of science to utilize their strengths in order to solve problems. In addition to that, the knowledge reuse in deep learning is an interesting aspect of this technology which will also be discussed.},
	language = {en},
	urldate = {2023-02-17},
	booktitle = {Deep {Learning}: {Convergence} to {Big} {Data} {Analytics}},
	publisher = {Springer},
	author = {Ahmad, Jamil and Farman, Haleem and Jan, Zahoor},
	editor = {Khan, Murad and Jan, Bilal and Farman, Haleem},
	year = {2019},
	doi = {10.1007/978-981-13-3459-7_3},
	pages = {31--42},
}

@inproceedings{nix_estimating_1994,
	title = {Estimating the mean and variance of the target probability distribution},
	volume = {1},
	doi = {10.1109/ICNN.1994.374138},
	abstract = {Introduces a method that estimates the mean and the variance of the probability distribution of the target as a function of the input, given an assumed target error-distribution model. Through the activation of an auxiliary output unit, this method provides a measure of the uncertainty of the usual network output for each input pattern. The authors derive the cost function and weight-update equations for the example of a Gaussian target error distribution, and demonstrate the feasibility of the network on a synthetic problem where the true input-dependent noise level is known.{\textbackslash}textless{\textbackslash}textgreater},
	booktitle = {Proceedings of 1994 {IEEE} {International} {Conference} on {Neural} {Networks} ({ICNN}'94)},
	author = {Nix, D.A. and Weigend, A.S.},
	month = jun,
	year = {1994},
	keywords = {Cognitive science, Computer errors, Computer science, Cost function, Equations, Error correction, Feedforward systems, Measurement uncertainty, Noise level, Probability distribution},
	pages = {55--60 vol.1},
}

@inproceedings{mathew_deep_2021,
	address = {Singapore},
	series = {Advances in {Intelligent} {Systems} and {Computing}},
	title = {Deep {Learning} {Techniques}: {An} {Overview}},
	isbn = {9789811533839},
	shorttitle = {Deep {Learning} {Techniques}},
	doi = {10.1007/978-981-15-3383-9_54},
	abstract = {Deep learning is a class of machine learning whichMathew, Amitha performsAmudha, P. muchSivakumari, S. better on unstructured data. Deep learning techniques are outperforming current machine learning techniques. It enables computational models to learn features progressively from data at multiple levels. The popularity of deep learning amplified as the amount of data available increased as well as the advancement of hardware that provides powerful computers. This article comprises the evolution of deep learning, various approaches to deep learning, architectures of deep learning, methods, and applications.},
	language = {en},
	booktitle = {Advanced {Machine} {Learning} {Technologies} and {Applications}},
	publisher = {Springer},
	author = {Mathew, Amitha and Amudha, P. and Sivakumari, S.},
	editor = {Hassanien, Aboul Ella and Bhatnagar, Roheet and Darwish, Ashraf},
	year = {2021},
	keywords = {Convolutional neural network (CNN), Deep belief network (DBN), Deep learning (DL), Generative adversarial network (GAN), Recurrent neural network (RNN)},
	pages = {599--608},
}

@misc{liu_simple_2020-2,
	title = {Simple and {Principled} {Uncertainty} {Estimation} with {Deterministic} {Deep} {Learning} via {Distance} {Awareness}},
	abstract = {Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer with a Gaussian process. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.},
	urldate = {2023-02-16},
	publisher = {arXiv},
	author = {Liu, Jeremiah Zhe and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax-Weiss, Tania and Lakshminarayanan, Balaji},
	month = oct,
	year = {2020},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@book{altland_condensed_2010,
	address = {Cambridge},
	edition = {2},
	title = {Condensed {Matter} {Field} {Theory}},
	isbn = {978-0-521-76975-4},
	abstract = {Modern experimental developments in condensed matter and ultracold atom physics present formidable challenges to theorists. This book provides a pedagogical introduction to quantum field theory in many-particle physics, emphasizing the applicability of the formalism to concrete problems. This second edition contains two new chapters developing path integral approaches to classical and quantum nonequilibrium phenomena. Other chapters cover a range of topics, from the introduction of many-body techniques and functional integration, to renormalization group methods, the theory of response functions, and topology. Conceptual aspects and formal methodology are emphasized, but the discussion focuses on practical experimental applications drawn largely from condensed matter physics and neighboring fields. Extended and challenging problems with fully worked solutions provide a bridge between formal manipulations and research-oriented thinking. Aimed at elevating graduate students to a level where they can engage in independent research, this book complements graduate level courses on many-particle theory.},
	urldate = {2023-02-16},
	publisher = {Cambridge University Press},
	author = {Altland, Alexander and Simons, Ben D.},
	year = {2010},
	doi = {10.1017/CBO9780511789984},
}

@article{altl_condensed_nodate,
	title = {Condensed {Matter} {Field} {Theory}, {Second} {Edition}},
	language = {en},
	journal = {Condensed Matter Field Theory},
	author = {Altl, Alex and er, Simons Ben D},
}

@article{altl_condensed_nodate-1,
	title = {Condensed {Matter} {Field} {Theory}, {Second} {Edition}},
	language = {en},
	journal = {Condensed Matter Field Theory},
	author = {Altl, Alex and er, Simons Ben D},
}

@misc{doyle_random_2000,
	title = {Random {Walks} and {Electric} {Networks}},
	abstract = {A popular account of the connection between random walks and electric networks.},
	urldate = {2023-02-16},
	publisher = {arXiv},
	author = {Doyle, Peter G. and Snell, J. Laurie},
	month = jan,
	year = {2000},
	keywords = {Mathematics - Probability},
}

@inproceedings{goldberg_regression_1997,
	title = {Regression with {Input}-dependent {Noise}: {A} {Gaussian} {Process} {Treatment}},
	volume = {10},
	shorttitle = {Regression with {Input}-dependent {Noise}},
	url = {https://proceedings.neurips.cc/paper/1997/hash/afe434653a898da20044041262b3ac74-Abstract.html},
	abstract = {Gaussian processes provide natural non-parametric prior distribu(cid:173) tions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both pro(cid:173) cesses can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.},
	urldate = {2023-02-11},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {MIT Press},
	author = {Goldberg, Paul and Williams, Christopher and Bishop, Christopher},
	year = {1997},
}

@book{rasmussen_gaussian_2006,
	address = {Cambridge, Mass},
	series = {Adaptive computation and machine learning},
	title = {Gaussian processes for machine learning},
	isbn = {978-0-262-18253-9},
	language = {en},
	publisher = {MIT Press},
	author = {Rasmussen, Carl Edward and Williams, Christopher K. I.},
	year = {2006},
	keywords = {Data processing, Gaussian processes, Machine learning, Mathematical models},
}

@misc{stirn_variational_2020,
	title = {Variational {Variance}: {Simple}, {Reliable}, {Calibrated} {Heteroscedastic} {Noise} {Variance} {Parameterization}},
	shorttitle = {Variational {Variance}},
	abstract = {Brittle optimization has been observed to adversely impact model likelihoods for regression and VAEs when simultaneously fitting neural network mappings from a (random) variable onto the mean and variance of a dependent Gaussian variable. Previous works have bolstered optimization and improved likelihoods, but fail other basic posterior predictive checks (PPCs). Under the PPC framework, we propose critiques to test predictive mean and variance calibration and the predictive distribution's ability to generate sensible data. We find that our attractively simple solution, to treat heteroscedastic variance variationally, sufficiently regularizes variance to pass these PPCs. We consider a diverse gamut of existing and novel priors and find our methods preserve or outperform existing model likelihoods while significantly improving parameter calibration and sample quality for regression and VAEs.},
	urldate = {2023-02-11},
	publisher = {arXiv},
	author = {Stirn, Andrew and Knowles, David A.},
	month = oct,
	year = {2020},
	doi = {10.48550/arXiv.2006.04910},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{guo_calibration_2017,
	title = {On {Calibration} of {Modern} {Neural} {Networks}},
	abstract = {Confidence calibration – the problem of predicting probability estimates representative of the true correctness likelihood – is important for classification models in many applications. We discover that modern neural networks, unlike those from a decade ago, are poorly calibrated. Through extensive experiments, we observe that depth, width, weight decay, and Batch Normalization are important factors influencing calibration. We evaluate the performance of various post-processing calibration methods on state-of-the-art architectures with image and document classification datasets. Our analysis and experiments not only offer insights into neural network learning, but also provide a simple and straightforward recipe for practical settings: on most datasets, temperature scaling – a single-parameter variant of Platt Scaling – is surprisingly effective at calibrating predictions.},
	urldate = {2023-02-10},
	publisher = {arXiv},
	author = {Guo, Chuan and Pleiss, Geoff and Sun, Yu and Weinberger, Kilian Q.},
	month = aug,
	year = {2017},
	doi = {10.48550/arXiv.1706.04599},
	keywords = {Computer Science - Machine Learning},
}

@inproceedings{skafte_reliable_2019-1,
	title = {Reliable training and estimation of variance networks},
	volume = {32},
	url = {https://proceedings.neurips.cc/paper/2019/hash/07211688a0869d995947a8fb11b215d6-Abstract.html},
	abstract = {We propose and investigate new complementary methodologies for estimating predictive variance networks in regression neural networks. We derive a locally aware mini-batching scheme that results in sparse robust gradients, and we show how to make unbiased weight updates to a variance network. Further, we formulate a heuristic for robustly fitting both the mean and variance networks post hoc. Finally, we take inspiration from posterior Gaussian processes and propose a network architecture with similar extrapolation properties to Gaussian processes. The proposed methodologies are complementary, and improve upon baseline methods individually. Experimentally, we investigate the impact of predictive uncertainty on multiple datasets and tasks ranging from regression, active learning and generative modeling. Experiments consistently show significant improvements in predictive uncertainty estimation over state-of-the-art methods across tasks and datasets.},
	urldate = {2023-02-02},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Skafte, Nicki and Jø rgensen, Martin and Hauberg, Sø ren},
	year = {2019},
}

@inproceedings{takahashi_student-t_2018,
	address = {Stockholm, Sweden},
	title = {Student-t {Variational} {Autoencoder} for {Robust} {Density} {Estimation}},
	isbn = {978-0-9992411-2-7},
	doi = {10.24963/ijcai.2018/374},
	abstract = {We propose a robust multivariate density estimator based on the variational autoencoder (VAE). The VAE is a powerful deep generative model, and used for multivariate density estimation. With the original VAE, the distribution of observed continuous variables is assumed to be a Gaussian, where its mean and variance are modeled by deep neural networks taking latent variables as their inputs. This distribution is called the decoder. However, the training of VAE often becomes unstable. One reason is that the decoder of VAE is sensitive to the error between the data point and its estimated mean when its estimated variance is almost zero. We solve this instability problem by making the decoder robust to the error using a Bayesian approach to the variance estimation: we set a prior for the variance of the Gaussian decoder, and marginalize it out analytically, which leads to proposing the Student-t VAE. Numerical experiments with various datasets show that training of the Student-t VAE is robust, and the Student-t VAE achieves high density estimation performance.},
	language = {en},
	urldate = {2023-02-02},
	booktitle = {Proceedings of the {Twenty}-{Seventh} {International} {Joint} {Conference} on {Artificial} {Intelligence}},
	publisher = {International Joint Conferences on Artificial Intelligence Organization},
	author = {Takahashi, Hiroshi and Iwata, Tomoharu and Yamanaka, Yuki and Yamada, Masanori and Yagi, Satoshi},
	month = jul,
	year = {2018},
	pages = {2696--2702},
}

@inproceedings{damianou_deep_2013,
	title = {Deep {Gaussian} {Processes}},
	abstract = {In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples.},
	language = {en},
	urldate = {2023-01-30},
	booktitle = {Proceedings of the {Sixteenth} {International} {Conference} on {Artificial} {Intelligence} and {Statistics}},
	publisher = {PMLR},
	author = {Damianou, Andreas and Lawrence, Neil D.},
	month = apr,
	year = {2013},
	pages = {207--215},
}

@article{schulz_tutorial_2018,
	title = {A tutorial on {Gaussian} process regression: {Modelling}, exploring, and exploiting functions},
	volume = {85},
	issn = {0022-2496},
	shorttitle = {A tutorial on {Gaussian} process regression},
	doi = {10.1016/j.jmp.2018.03.001},
	abstract = {This tutorial introduces the reader to Gaussian process regression as an expressive tool to model, actively explore and exploit unknown functions. Gaussian process regression is a powerful, non-parametric Bayesian approach towards regression problems that can be utilized in exploration and exploitation scenarios. This tutorial aims to provide an accessible introduction to these techniques. We will introduce Gaussian processes which generate distributions over functions used for Bayesian non-parametric regression, and demonstrate their use in applications and didactic examples including simple regression problems, a demonstration of kernel-encoded prior assumptions and compositions, a pure exploration scenario within an optimal design framework, and a bandit-like exploration–exploitation scenario where the goal is to recommend movies. Beyond that, we describe a situation modelling risk-averse exploration in which an additional constraint (not to sample below a certain threshold) needs to be accounted for. Lastly, we summarize recent psychological experiments utilizing Gaussian processes. Software and literature pointers are also provided.},
	language = {en},
	urldate = {2023-01-30},
	journal = {Journal of Mathematical Psychology},
	author = {Schulz, Eric and Speekenbrink, Maarten and Krause, Andreas},
	month = aug,
	year = {2018},
	keywords = {Active learning, Bandit problems, Exploration–exploitation, Gaussian process regression},
	pages = {1--16},
}

@inproceedings{le_heteroscedastic_2005,
	address = {Bonn, Germany},
	title = {Heteroscedastic {Gaussian} process regression},
	isbn = {978-1-59593-180-1},
	doi = {10.1145/1102351.1102413},
	abstract = {This paper presents an algorithm to estimate simultaneously both mean and variance of a non parametric regression problem. The key point is that we are able to estimate variance locally unlike standard Gaussian Process regression or SVMs. This means that our estimator adapts to the local noise. The problem is cast in the setting of maximum a posteriori estimation in exponential families. Unlike previous work, we obtain a convex optimization problem which can be solved via Newton’s method.},
	language = {en},
	booktitle = {Proceedings of the 22nd international conference on {Machine} learning - {ICML} '05},
	publisher = {ACM Press},
	author = {Le, Quoc V. and Smola, Alex J. and Canu, Stéphane},
	year = {2005},
	pages = {489--496},
}

@article{hinne_conceptual_2020,
	title = {A {Conceptual} {Introduction} to {Bayesian} {Model} {Averaging}},
	volume = {3},
	issn = {2515-2459},
	doi = {10.1177/2515245919898657},
	abstract = {Many statistical scenarios initially involve several candidate models that describe the data-generating process. Analysis often proceeds by first selecting the best model according to some criterion and then learning about the parameters of this selected model. Crucially, however, in this approach the parameter estimates are conditioned on the selected model, and any uncertainty about the model-selection process is ignored. An alternative is to learn the parameters for all candidate models and then combine the estimates according to the posterior probabilities of the associated models. This approach is known as Bayesian model averaging (BMA). BMA has several important advantages over all-or-none selection methods, but has been used only sparingly in the social sciences. In this conceptual introduction, we explain the principles of BMA, describe its advantages over all-or-none model selection, and showcase its utility in three examples: analysis of covariance, meta-analysis, and network analysis.},
	number = {2},
	journal = {Advances in Methods and Practices in Psychological Science},
	author = {Hinne, Max and Gronau, Quentin F. and van den Bergh, Don and Wagenmakers, Eric-Jan},
	month = jun,
	year = {2020},
	pages = {200--215},
}

@misc{collier_simple_2020,
	title = {A {Simple} {Probabilistic} {Method} for {Deep} {Classification} under {Input}-{Dependent} {Label} {Noise}},
	abstract = {Datasets with noisy labels are a common occurrence in practical applications of classification methods. We propose a simple probabilistic method for training deep classifiers under input-dependent (heteroscedastic) label noise. We assume an underlying heteroscedastic generative process for noisy labels. To make gradient based training feasible we use a temperature parameterized softmax as a smooth approximation to the assumed generative process. We illustrate that the softmax temperature controls a bias-variance trade-off for the approximation. By tuning the softmax temperature, we improve accuracy, log-likelihood and calibration on both image classification benchmarks with controlled label noise as well as Imagenet-21k which has naturally occurring label noise. For image segmentation, our method increases the mean IoU on the PASCAL VOC and Cityscapes datasets by more than 1\% over the state-of-the-art model.},
	urldate = {2023-01-30},
	publisher = {arXiv},
	author = {Collier, Mark and Mustafa, Basil and Kokiopoulou, Efi and Jenatton, Rodolphe and Berent, Jesse},
	month = nov,
	year = {2020},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{noauthor_variational_2022,
	title = {Variational {Variance}: {Simple} and {Reliable} {Noise} {Variance} {Parameterization}},
	shorttitle = {Variational {Variance}},
	abstract = {Models employing heteroscedastic Gaussian likelihoods parameterized by amortized mean and variance networks are both probabilistically interpretable and highly flexible, but unfortunately can be brittle to optimize. Maximizing log likelihood encourages local Dirac densities for sufficiently flexible mean and variance networks. Data lacking nearby neighbors can provide this flexibility. Gradients near these unbounded optima explode, prohibiting convergence of the mean and thus requiring high noise variance to explain the dependent variable. We propose posterior predictive checks to identify such failures, which we observe can surreptitiously occur alongside high model likelihoods. We find existing approaches that bolster optimization of mean and variance networks to improve likelihoods still exhibit poor predictive mean and variance calibrations. Our notably simpler solution, to treat heteroscedastic variance variationally in an Empirical Bayes regime, regularizes variance away from zero and stabilizes optimization, allowing us to preserve or outperform existing likelihoods while improving predictive mean and variance calibrations and thereby sample quality. We empirically demonstrate these findings on a variety of regression and variational autoencoding tasks.},
	language = {en},
	urldate = {2023-01-30},
	journal = {Transactions on Machine Learning Research},
	month = sep,
	year = {2022},
}

@article{abdar_review_2021,
	title = {A review of uncertainty quantification in deep learning: {Techniques}, applications and challenges},
	volume = {76},
	issn = {1566-2535},
	shorttitle = {A review of uncertainty quantification in deep learning},
	doi = {10.1016/j.inffus.2021.05.008},
	abstract = {Uncertainty quantification (UQ) methods play a pivotal role in reducing the impact of uncertainties during both optimization and decision making processes. They have been applied to solve a variety of real-world problems in science and engineering. Bayesian approximation and ensemble learning techniques are two widely-used types of uncertainty quantification (UQ) methods. In this regard, researchers have proposed different UQ methods and examined their performance in a variety of applications such as computer vision (e.g., self-driving cars and object detection), image processing (e.g., image restoration), medical image analysis (e.g., medical image classification and segmentation), natural language processing (e.g., text classification, social media texts and recidivism risk-scoring), bioinformatics, etc. This study reviews recent advances in UQ methods used in deep learning, investigates the application of these methods in reinforcement learning, and highlights fundamental research challenges and directions associated with UQ.},
	language = {en},
	journal = {Information Fusion},
	author = {Abdar, Moloud and Pourpanah, Farhad and Hussain, Sadiq and Rezazadegan, Dana and Liu, Li and Ghavamzadeh, Mohammad and Fieguth, Paul and Cao, Xiaochun and Khosravi, Abbas and Acharya, U. Rajendra and Makarenkov, Vladimir and Nahavandi, Saeid},
	month = dec,
	year = {2021},
	keywords = {Artificial intelligence, Bayesian statistics, Deep learning, Ensemble learning, Machine learning, Uncertainty quantification},
	pages = {243--297},
}

@inproceedings{kendall_what_2017,
	title = {What {Uncertainties} {Do} {We} {Need} in {Bayesian} {Deep} {Learning} for {Computer} {Vision}?},
	volume = {30},
	abstract = {There are two major types of uncertainty one can model. Aleatoric uncertainty captures noise inherent in the observations. On the other hand, epistemic uncertainty accounts for uncertainty in the model - uncertainty which can be explained away given enough data. Traditionally it has been difficult to model epistemic uncertainty in computer vision, but with new Bayesian deep learning tools this is now possible. We study the benefits of modeling epistemic vs. aleatoric uncertainty in Bayesian deep learning models for vision tasks. For this we present a Bayesian deep learning framework combining input-dependent aleatoric uncertainty together with epistemic uncertainty. We study models under the framework with per-pixel semantic segmentation and depth regression tasks. Further, our explicit uncertainty formulation leads to new loss functions for these tasks, which can be interpreted as learned attenuation. This makes the loss more robust to noisy data, also giving new state-of-the-art results on segmentation and depth regression benchmarks.},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Kendall, Alex and Gal, Yarin},
	year = {2017},
}

@misc{collier_simple_2020-1,
	title = {A {Simple} {Probabilistic} {Method} for {Deep} {Classification} under {Input}-{Dependent} {Label} {Noise}},
	abstract = {Datasets with noisy labels are a common occurrence in practical applications of classification methods. We propose a simple probabilistic method for training deep classifiers under input-dependent (heteroscedastic) label noise. We assume an underlying heteroscedastic generative process for noisy labels. To make gradient based training feasible we use a temperature parameterized softmax as a smooth approximation to the assumed generative process. We illustrate that the softmax temperature controls a bias-variance trade-off for the approximation. By tuning the softmax temperature, we improve accuracy, log-likelihood and calibration on both image classification benchmarks with controlled label noise as well as Imagenet-21k which has naturally occurring label noise. For image segmentation, our method increases the mean IoU on the PASCAL VOC and Cityscapes datasets by more than 1\% over the state-of-the-art model.},
	urldate = {2023-01-30},
	publisher = {arXiv},
	author = {Collier, Mark and Mustafa, Basil and Kokiopoulou, Efi and Jenatton, Rodolphe and Berent, Jesse},
	month = nov,
	year = {2020},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{khosla_neural_2022,
	title = {Neural {Active} {Learning} on {Heteroskedastic} {Distributions}},
	abstract = {Models that can actively seek out the best quality training data hold the promise of more accurate, adaptable, and efficient machine learning. State-of-the-art active learning techniques tend to prefer examples that are the most difficult to classify. While this works well on homogeneous datasets, we find that it can lead to catastrophic failures when performed on multiple distributions with different degrees of label noise or heteroskedasticity. These active learning algorithms strongly prefer to draw from the distribution with more noise, even if their examples have no informative structure (such as solid color images with random labels). To this end, we demonstrate the catastrophic failure of these active learning algorithms on heteroskedastic distributions and propose a fine-tuning-based approach to mitigate these failures. Further, we propose a new algorithm that incorporates a model difference scoring function for each data point to filter out the noisy examples and sample clean examples that maximize accuracy, outperforming the existing active learning techniques on the heteroskedastic datasets. We hope these observations and techniques are immediately helpful to practitioners and can help to challenge common assumptions in the design of active learning algorithms.},
	publisher = {arXiv},
	author = {Khosla, Savya and Whye, Chew Kin and Ash, Jordan T. and Zhang, Cyril and Kawaguchi, Kenji and Lamb, Alex},
	month = nov,
	year = {2022},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning},
}

@misc{cao_heteroskedastic_2021,
	title = {Heteroskedastic and {Imbalanced} {Deep} {Learning} with {Adaptive} {Regularization}},
	abstract = {Real-world large-scale datasets are heteroskedastic and imbalanced – labels have varying levels of uncertainty and label distributions are long-tailed. Heteroskedasticity and imbalance challenge deep learning algorithms due to the difficulty of distinguishing among mislabeled, ambiguous, and rare examples. Addressing heteroskedasticity and imbalance simultaneously is under-explored. We propose a data-dependent regularization technique for heteroskedastic datasets that regularizes different regions of the input space differently. Inspired by the theoretical derivation of the optimal regularization strength in a one-dimensional nonparametric classification setting, our approach adaptively regularizes the data points in higher-uncertainty, lower-density regions more heavily. We test our method on several benchmark tasks, including a real-world heteroskedastic and imbalanced dataset, WebVision. Our experiments corroborate our theory and demonstrate a significant improvement over other methods in noise-robust deep learning.},
	publisher = {arXiv},
	author = {Cao, Kaidi and Chen, Yining and Lu, Junwei and Arechiga, Nikos and Gaidon, Adrien and Ma, Tengyu},
	month = mar,
	year = {2021},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{hullermeier_aleatoric_2021,
	title = {Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods},
	volume = {110},
	issn = {1573-0565},
	shorttitle = {Aleatoric and epistemic uncertainty in machine learning},
	doi = {10.1007/s10994-021-05946-3},
	abstract = {The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often referred to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of attempts so far at handling uncertainty in general and formalizing this distinction in particular.},
	language = {en},
	number = {3},
	journal = {Machine Learning},
	author = {Hüllermeier, Eyke and Waegeman, Willem},
	month = mar,
	year = {2021},
	pages = {457--506},
}

@misc{lakshminarayanan_simple_2017-1,
	title = {Simple and {Scalable} {Predictive} {Uncertainty} {Estimation} using {Deep} {Ensembles}},
	url = {http://arxiv.org/abs/1612.01474},
	abstract = {Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs, which learn a distribution over weights, are currently the state-of-the-art for estimating predictive uncertainty; however these require significant modifications to the training procedure and are computationally expensive compared to standard (non-Bayesian) NNs. We propose an alternative to Bayesian NNs that is simple to implement, readily parallelizable, requires very little hyperparameter tuning, and yields high quality predictive uncertainty estimates. Through a series of experiments on classification and regression benchmarks, we demonstrate that our method produces well-calibrated uncertainty estimates which are as good or better than approximate Bayesian NNs. To assess robustness to dataset shift, we evaluate the predictive uncertainty on test examples from known and unknown distributions, and show that our method is able to express higher uncertainty on out-of-distribution examples. We demonstrate the scalability of our method by evaluating predictive uncertainty estimates on ImageNet.},
	publisher = {arXiv},
	author = {Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles},
	month = nov,
	year = {2017},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{wang_robust_2018,
	title = {Robust {Probabilistic} {Modeling} with {Bayesian} {Data} {Reweighting}},
	url = {http://arxiv.org/abs/1606.03860},
	abstract = {Probabilistic models analyze data by relying on a set of assumptions. Data that exhibit deviations from these assumptions can undermine inference and prediction quality. Robust models offer protection against mismatch between a model's assumptions and reality. We propose a way to systematically detect and mitigate mismatch of a large class of probabilistic models. The idea is to raise the likelihood of each observation to a weight and then to infer both the latent variables and the weights from data. Inferring the weights allows a model to identify observations that match its assumptions and down-weight others. This enables robust inference and improves predictive accuracy. We study four different forms of mismatch with reality, ranging from missing latent groups to structure misspecification. A Poisson factorization analysis of the Movielens 1M dataset shows the benefits of this approach in a practical scenario.},
	publisher = {arXiv},
	author = {Wang, Yixin and Kucukelbir, Alp and Blei, David M.},
	month = jun,
	year = {2018},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{uyanto_monte_2022,
	title = {Monte {Carlo} power comparison of seven most commonly used heteroscedasticity tests},
	volume = {51},
	issn = {0361-0918},
	doi = {10.1080/03610918.2019.1692031},
	abstract = {Assumption of the classical linear regression model states that the disturbances should have a constant (equal) variance. When this requirement is not met, the loss in efficiency in using ordinary least squares may be substantial and the biases in estimated standard errors may lead to invalid inferences. This problem is known as heteroscedasticity. There are many tests for heteroscedasticity, we want to know which test is more powerful. We use Monte Carlo simulation to compare the power of seven most commonly used tests for detecting heteroscedasticity, namely, Breusch–Pagan test, Glejser test, Goldfeld–Quandt test, Harvey–Godfrey test, Harrison–McCabe test, Park test, and White test for six common types of heteroscedasticity. Simulation results show that the Harrison–McCabe test has generally the most power in all of the six common types of heteroscedasticity and the White test has generally the least power in all of the six common types of heteroscedasticity.},
	number = {4},
	journal = {Communications in Statistics - Simulation and Computation},
	author = {Uyanto, Stanislaus S.},
	month = apr,
	year = {2022},
	keywords = {Homocedasticity, Ordinary least squares, Regression diagnostic, Residuals},
	pages = {2065--2082},
}

@article{pratola_heteroscedastic_2020,
	title = {Heteroscedastic {BART} via {Multiplicative} {Regression} {Trees}},
	volume = {29},
	issn = {1061-8600},
	doi = {10.1080/10618600.2019.1677243},
	abstract = {Bayesian additive regression trees (BART) has become increasingly popular as a flexible and scalable nonparametric regression approach for modern applied statistics problems. For the practitioner dealing with large and complex nonlinear response surfaces, its advantages include a matrix-free formulation and the lack of a requirement to prespecify a confining regression basis. Although flexible in fitting the mean, BART has been limited by its reliance on a constant variance error model. Alleviating this limitation, we propose HBART, a nonparametric heteroscedastic elaboration of BART. In BART, the mean function is modeled with a sum of trees, each of which determines an additive contribution to the mean. In HBART, the variance function is further modeled with a product of trees, each of which determines a multiplicative contribution to the variance. Like the mean model, this flexible, multidimensional variance model is entirely nonparametric with no need for the prespecification of a confining basis. Moreover, with this enhancement, HBART can provide insights into the potential relationships of the predictors with both the mean and the variance. Practical implementations of HBART with revealing new diagnostic plots are demonstrated with simulated and real data on used car prices and song year of release. Supplementary materials for this article are available online.},
	number = {2},
	journal = {Journal of Computational and Graphical Statistics},
	author = {Pratola, M. T. and Chipman, H. A. and George, E. I. and McCulloch, R. E.},
	month = apr,
	year = {2020},
	keywords = {Applied statistical inference, Big data, Nonparametric regression, Uncertainty quantification},
	pages = {405--417},
}

@article{parker_general_2021,
	title = {A general {Bayesian} model for heteroskedastic data with fully conjugate full-conditional distributions},
	volume = {91},
	issn = {0094-9655},
	doi = {10.1080/00949655.2021.1925279},
	abstract = {Models for heteroskedastic data are relevant in a variety of applications ranging from financial time series to environmental statistics. However, the topic of modelling the variance function conditionally has not seen as much attention as modelling the mean. Volatility models have been used in specific applications, but these models can be difficult to fit in a Bayesian setting due to posterior distributions that are challenging to sample efficiently. In this work, we introduce a general model for heteroskedastic data. This approach models the conditional variance as a function of any desired covariates or random effects. We rely on multivariate log-Gamma distribution theory to construct priors that yield fully conjugate full-conditional distributions for Gibbs sampling. Furthermore, we extend the model to a deep learning approach that can provide highly accurate estimates for time dependent data. We also provide an extension for heavy-tailed data. We illustrate our methodology via three applications.},
	number = {15},
	journal = {Journal of Statistical Computation and Simulation},
	author = {Parker, Paul A. and Holan, Scott H. and Wills, Skye A.},
	month = oct,
	year = {2021},
	keywords = {Deep learning, Gibbs sampling, echo state network, mixed models, multivariate log-Gamma, spatial, volatility},
	pages = {3207--3227},
}

@article{eubank_detecting_1993,
	title = {Detecting {Heteroscedasticity} in {Nonparametric} {Regression}},
	volume = {55},
	issn = {2517-6161},
	doi = {10.1111/j.2517-6161.1993.tb01474.x},
	abstract = {Diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression. The large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances. New diagnostic plots are also developed and illustrated.},
	language = {en},
	number = {1},
	journal = {Journal of the Royal Statistical Society: Series B (Methodological)},
	author = {Eubank, R. L. and Thomas, Will},
	year = {1993},
	keywords = {asymptotic distribution, diagnostic plots, scores test},
	pages = {145--155},
}

@article{chesher_finite-sample_1991,
	title = {The finite-sample distributions of heteroskedasticity robust {Wald} statistics},
	volume = {47},
	issn = {0304-4076},
	doi = {10.1016/0304-4076(91)90082-O},
	abstract = {The Imhof procedure is used to calculate the exact finite-sample distributions of alternative heteroskedasticity robust Wald-type tests of scalar linear hypothesis in the normal linear model. The tests are distinguished by the variance estimator that they use. These include the White-Eicker and jackknife estimators. The calculations demonstrate the importance of regression design in determining the quality of conventional first-order asymptotic approximations to the null distributions of the tests.},
	language = {en},
	number = {1},
	journal = {Journal of Econometrics},
	author = {Chesher, Andrew and Austin, Gerard},
	month = jan,
	year = {1991},
	pages = {153--173},
}

@misc{seitzer_pitfalls_2022-1,
	title = {On the {Pitfalls} of {Heteroscedastic} {Uncertainty} {Estimation} with {Probabilistic} {Neural} {Networks}},
	url = {http://arxiv.org/abs/2203.09168},
	abstract = {Capturing aleatoric uncertainty is a critical part of many machine learning systems. In deep learning, a common approach to this end is to train a neural network to estimate the parameters of a heteroscedastic Gaussian distribution by maximizing the logarithm of the likelihood function under the observed data. In this work, we examine this approach and identify potential hazards associated with the use of log-likelihood in conjunction with gradient-based optimizers. First, we present a synthetic example illustrating how this approach can lead to very poor but stable parameter estimates. Second, we identify the culprit to be the log-likelihood loss, along with certain conditions that exacerbate the issue. Third, we present an alternative formulation, termed \{\vphantom{\}}{\textbackslash}textbackslashbeta{\textbackslash}-NLL, in which each data point's contribution to the loss is weighted by the \{\vphantom{\}}{\textbackslash}textbackslashbeta{\textbackslash}-exponentiated variance estimate. We show that using an appropriate \{\vphantom{\}}{\textbackslash}textbackslashbeta{\textbackslash} largely mitigates the issue in our illustrative example. Fourth, we evaluate this approach on a range of domains and tasks and show that it achieves considerable improvements and performs more robustly concerning hyperparameters, both in predictive RMSE and log-likelihood criteria.},
	publisher = {arXiv},
	author = {Seitzer, Maximilian and Tavakoli, Arash and Antic, Dimitrije and Martius, Georg},
	month = apr,
	year = {2022},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{huber_behavior_1967,
	title = {The behavior of maximum likelihood estimates under nonstandard conditions},
	volume = {5.1},
	abstract = {Berkeley Symposium on Mathematical Statistics and Probability},
	journal = {Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Statistics},
	author = {Huber, Peter J.},
	month = jan,
	year = {1967},
	pages = {221--234},
}

@article{stekhoven_missforestnon-parametric_2011,
	title = {{MissForest}—non-parametric missing value imputation for mixed-type data},
	volume = {28},
	issn = {1367-4803},
	url = {https://doi.org/10.1093/bioinformatics/btr597},
	doi = {10.1093/bioinformatics/btr597},
	abstract = {Motivation: Modern data acquisition based on high-throughput technology is often facing the problem of missing data. Algorithms commonly used in the analysis of such large-scale data often depend on a complete set. Missing value imputation offers a solution to this problem. However, the majority of available imputation methods are restricted to one type of variable only: continuous or categorical. For mixed-type data, the different types are usually handled separately. Therefore, these methods ignore possible relations between variable types. We propose a non-parametric method which can cope with different types of variables simultaneously.Results: We compare several state of the art methods for the imputation of missing values. We propose and evaluate an iterative imputation method (missForest) based on a random forest. By averaging over many unpruned classification or regression trees, random forest intrinsically constitutes a multiple imputation scheme. Using the built-in out-of-bag error estimates of random forest, we are able to estimate the imputation error without the need of a test set. Evaluation is performed on multiple datasets coming from a diverse selection of biological fields with artificially introduced missing values ranging from 10\% to 30\%. We show that missForest can successfully handle missing values, particularly in datasets including different types of variables. In our comparative study, missForest outperforms other methods of imputation especially in data settings where complex interactions and non-linear relations are suspected. The out-of-bag imputation error estimates of missForest prove to be adequate in all settings. Additionally, missForest exhibits attractive computational efficiency and can cope with high-dimensional data.Availability: The ℝ package missForest is freely available from http://stat.ethz.ch/CRAN/.Contact:stekhoven@stat.math.ethz.ch; buhlmann@stat.math.ethz.ch},
	number = {1},
	journal = {Bioinformatics},
	author = {Stekhoven, Daniel J. and Bühlmann, Peter},
	month = oct,
	year = {2011},
	pages = {112--118},
}

@article{hahn-holbrook_placental_2013,
	title = {Placental {Corticotropin}-{Releasing} {Hormone} {Mediates} the {Association} {Between} {Prenatal} {Social} {Support} and {Postpartum} {Depression}},
	volume = {1},
	issn = {2167-7026},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3756599/},
	doi = {10.1177/2167702612470646},
	abstract = {Three decades of research point to both biological and psychological risk factors for postpartum depression, but very little research integrates the two. This study bridged this gap by testing whether prenatal social support predicted depressive symptoms at 8 weeks postpartum in a multiethnic sample of 210 women and whether the stress hormone placental corticotropin-releasing hormone (pCRH), measured at 19, 29, and 37 weeks’ gestation, mediated this relationship. We found that prenatal family support predicted significantly fewer depressive symptoms postpartum and more gradual increases in pCRH from 29 to 37 weeks’ gestation. Furthermore, steeper increases in pCRH during this same period predicted more depressive symptoms postpartum. Finally, these changes in pCRH in late pregnancy mediated the relationship between prenatal family support and postpartum depressive symptoms. These results suggest that social and biological risk factors for postpartum depressive symptoms are intertwined and move us closer to an integrated biopsychosocial understanding of postpartum depression.},
	number = {3},
	urldate = {2022-04-18},
	journal = {Clinical psychological science : \$b a journal of the Association for Psychological Science},
	author = {Hahn-Holbrook, Jennifer and Schetter, Christine Dunkel and Arora, Chander and Hobel, Calvin J.},
	month = jul,
	year = {2013},
	pmid = {23997996},
	pmcid = {PMC3756599},
	pages = {253--264},
}

@article{wong-toi_joint_2022,
	title = {A {Joint} {Analysis} for {Field} {Goal} {Attempts} and {Percentages} of {Professional} {Basketball} {Players}: {Bayesian} {Nonparametric} {Resource}},
	issn = {1680-743X},
	doi = {10.6339/22-JDS1062},
	journal = {Journal of Data Science},
	author = {Wong-Toi, Eliot and Yang, Hou-Cheng and Shen, Weining and Hu, Guanyu},
	year = {2022},
	pages = {1--19},
}

@article{gaillard_maternal_2014,
	title = {Maternal {Parity}, {Fetal} and {Childhood} {Growth}, and {Cardiometabolic} {Risk} {Factors}},
	volume = {64},
	url = {https://www.ahajournals.org/doi/10.1161/hypertensionaha.114.03492},
	doi = {10.1161/HYPERTENSIONAHA.114.03492},
	abstract = {We examined the associations of maternal parity with fetal and childhood growth characteristics and childhood cardiometabolic risk factors in a population-based prospective cohort study among 9031 mothers and their children. Fetal and childhood growth were repeatedly measured. We measured childhood anthropometrics, body fat distribution, left ventricular mass, blood pressure, blood lipids, and insulin levels at the age of 6 years. Compared with nulliparous mothers, multiparous mothers had children with higher third trimester fetal head circumference, length and weight growth, and lower risks of preterm birth and small-size-for-gestational-age at birth but a higher risk of large-size-for-gestational-age at birth (P{\textbackslash}textless0.05). Children from multiparous mothers had lower rates of accelerated infant growth and lower levels of childhood body mass index, total fat mass percentage, and total and low-density lipoprotein cholesterol than children of nulliparous mothers (P{\textbackslash}textless0.05). They also had a lower risk of childhood overweight (odds ratio, 0.75 [95\% confidence interval, 0.63–0.88]). The risk of childhood clustering of cardiometabolic risk factors was not statistically significantly different (odds ratio, 0.82; 95\% confidence interval, 0.64–1.05). Among children from multiparous mothers only, we observed consistent trends toward a lower risk of childhood overweight and lower cholesterol levels with increasing parity (P{\textbackslash}textless0.05). In conclusion, offspring from nulliparous mothers have lower fetal but higher infant growth rates and higher risks of childhood overweight and adverse metabolic profile. Maternal nulliparity may have persistent cardiometabolic consequences for the offspring.},
	number = {2},
	urldate = {2022-04-18},
	journal = {Hypertension},
	author = {Gaillard, Romy and Rurangirwa, Akashi A. and Williams, Michelle A. and Hofman, Albert and Mackenbach, Johan P. and Franco, Oscar H. and Steegers, Eric A.P. and Jaddoe, Vincent W.V.},
	month = aug,
	year = {2014},
	keywords = {fetal development, parity, pediatric obesity},
	pages = {266--274},
}

@article{bai_parity_2002,
	title = {Parity and pregnancy outcomes},
	volume = {186},
	issn = {0002-9378, 1097-6868},
	url = {https://www.ajog.org/article/S0002-9378(02)46017-6/fulltext},
	doi = {10.1067/mob.2002.119639},
	language = {English},
	number = {2},
	urldate = {2022-04-18},
	journal = {American Journal of Obstetrics \& Gynecology},
	author = {Bai, Jun and Wong, Felix W. S. and Bauman, Adrian and Mohsin, Mohammed},
	month = feb,
	year = {2002},
	keywords = {Parity, neonatal morbidity, obstetric complications, perinatal death, pregnancy outcomes},
	pages = {274--278},
}

@article{nuttall_body_2015,
	title = {Body {Mass} {Index}: {Obesity}, {BMI}, and {Health}: {A} {Critical} {Review}},
	volume = {50},
	issn = {0029-666X},
	shorttitle = {Body {Mass} {Index}},
	url = {https://journals.lww.com/nutritiontodayonline/fulltext/2015/05000/body_mass_index__obesity,_bmi,_and_health__a.5.aspx},
	doi = {10.1097/NT.0000000000000092},
	abstract = {The body mass index (BMI) is the metric currently in use for defining anthropometric height/weight characteristics in adults and for classifying (categorizing) them into groups. The common interpretation is that it represents an index of an individual’s fatness. It also is widely used as a risk factor for the development of or the prevalence of several health issues. In addition, it is widely used in determining public health policies.The BMI has been useful in population-based studies by virtue of its wide acceptance in defining specific categories of body mass as a health issue. However, it is increasingly clear that BMI is a rather poor indicator of percent of body fat. Importantly, the BMI also does not capture information on the mass of fat in different body sites. The latter is related not only to untoward health issues but to social issues as well. Lastly, current evidence indicates there is a wide range of BMIs over which mortality risk is modest, and this is age related. All of these issues are discussed in this brief review.},
	language = {en-US},
	number = {3},
	urldate = {2022-04-18},
	journal = {Nutrition Today},
	author = {Nuttall, Frank Q.},
	month = jun,
	year = {2015},
	pages = {117--128},
}

@article{noauthor_national_nodate,
	title = {National {Health} and {Nutrition} {Examination} {Survey} (7/03)},
	language = {en},
	pages = {2},
}

@misc{noauthor_zotero_nodate,
	title = {Zotero {\textbackslash}textbar {Email} validation},
	url = {https://www.zotero.org/user/validate/bfe5427c8b14edad3d93},
	urldate = {2020-06-22},
}

@misc{noauthor_terms_2019,
	title = {Terms, {Definitions}, and {Calculations} {\textbackslash}textbar {Surveillance} {Overview} {\textbackslash}textbar {Statistics} {Center} {\textbackslash}textbar {HIV}/{AIDS} {\textbackslash}textbar {CDC}},
	url = {https://www.cdc.gov/hiv/statistics/surveillance/terms.html},
	abstract = {Surveillance Brief: Terms, Definitions, and Calculations Used in CDC HIV Surveillance Publications.},
	language = {en-us},
	urldate = {2020-06-23},
	month = oct,
	year = {2019},
}

@misc{noauthor_understanding_2020,
	title = {Understanding {Care} {\textbackslash}textbar {Living} {With} {HIV} {\textbackslash}textbar {HIV} {Basics} {\textbackslash}textbar {HIV}/{AIDS} {\textbackslash}textbar {CDC}},
	url = {https://www.cdc.gov/hiv/basics/livingwithhiv/understanding-care.html},
	abstract = {Support for staying in care and on HIV treatment.},
	language = {en-us},
	urldate = {2020-06-23},
	month = jun,
	year = {2020},
}

@misc{noauthor_zidovudine_nodate,
	title = {Zidovudine: {MedlinePlus} {Drug} {Information}},
	shorttitle = {Zidovudine},
	url = {https://medlineplus.gov/druginfo/meds/a687007.html},
	abstract = {Zidovudine: learn about side effects, dosage, special precautions, and more on MedlinePlus},
	language = {en},
	urldate = {2020-06-23},
}

@misc{noauthor_natural_nodate,
	title = {The natural history and clinical features of {HIV} infection in adults and adolescents - {UpToDate}},
	url = {https://www.uptodate.com/contents/the-natural-history-and-clinical-features-of-hiv-infection-in-adults-and-adolescents?search=cd4&source=search_result&selectedTitle=5~150&usage_type=default&display_rank=5#H4},
	urldate = {2020-06-23},
}

@article{raboud_quantification_1995,
	title = {Quantification of the variation due to laboratory and physiologic sources in {CD4} lymphocyte counts of clinically stable {HIV}-infected individuals},
	volume = {10 Suppl 2},
	issn = {1077-9450},
	abstract = {We have conducted a study to quantify the amount of variation in the CD4 lymphocyte counts of HIV-infected individuals due to laboratory and physiological factors. Thirty HIV-infected male volunteers had blood drawn on six occasions: three times in each of 2 weeks, 4 weeks apart. Two tubes of blood were drawn at each visit, and duplicate measurements were obtained from one of the tubes of blood. Differences between duplicate measurements from a single tube of blood and between CD4 counts obtained from two tubes of blood drawn on the same day were attributed to laboratory factors. Differences between CD4 counts obtained on different days were attributed to a combination of laboratory factors and physiologic factors, which included the effects of exercise, tobacco, and the consumption of alcohol and caffeine. The mean absolute CD4 count at the first visit was 450 (range 86-1,081). The short-term coefficient of variation of CD4 count was 13.7 (95\% CI: 12.9, 14.6). Physiologic and laboratory factors accounted for 85\% and 15\% of the variation in CD4 counts, respectively. Variation in the absolute white blood cell count, lymphocyte percentage, and CD4 percentage accounted fo 52\%, 29\%, and 19\% of the physiologic variation in CD4 counts, respectively. Our results confirm a high degree of short-term variability of CD4 counts among HIV-infected individuals, which can be largely attributed to physiological factors. This variability can be minimized more effectively by repeating CD4 counts over time than by repeating measurements at a single visit.(ABSTRACT TRUNCATED AT 250 WORDS)},
	language = {eng},
	journal = {Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology: Official Publication of the International Retrovirology Association},
	author = {Raboud, J. M. and Haley, L. and Montaner, J. S. and Murphy, C. and Januszewska, M. and Schechter, M. T.},
	year = {1995},
	pmid = {7552516},
	keywords = {Adult, Alcohol Drinking, Antiviral Agents, Biomarkers, CD4 Lymphocyte Count, Caffeine, Exercise, HIV Infections, Humans, Male, Middle Aged, Reproducibility of Results, Smoking, variation},
	pages = {S67--73},
}

@article{prins_gender_1999,
	title = {Do gender differences in {CD4} cell counts matter?},
	volume = {13},
	issn = {0269-9370},
	doi = {10.1097/00002030-199912030-00007},
	abstract = {OBJECTIVE: To examine the effect of gender on disease progression and whether gender differences in CD4 lymphocyte counts persisted for the entire course from HIV seroconversion until (death from) AIDS. METHODS: CD4 lymphocyte counts were modelled in 221 female and 443 male seroconverters following seroconversion, backwards from AIDS and backwards from death using regression analysis for repeated measurements. RESULTS: In the period before use of highly active antiretroviral therapy (HAART), progression to AIDS and to death were marginally slower in women than in men as assessed by proportional hazards analysis. Women seroconverted for HIV, developed AIDS and died at higher CD4 cell counts than men (women: 815, 146 and 44 x 10(6) cells/l, respectively; men: 727, 49 and 22 x 10(6) cells/l, respectively), although differences were only statistically significant at AIDS onset. Declines in CD4 lymphocyte counts were not significantly affected by gender and absolute differences between men and women were stable, with exception for the trajectory close to AIDS when the decline became steeper for men than women. CONCLUSION: These gender differences in CD4 lymphocyte counts suggest a delay of initiation of therapy in women compared with men (our model predicted that women reach the threshold of starting HAART at about 12 months later than men). If this delay unfavourably influences progression, treatment guidelines should be revised so that women can benefit equally from HAART.},
	language = {eng},
	number = {17},
	journal = {AIDS (London, England)},
	author = {Prins, M. and Robertson, J. R. and Brettle, R. P. and Aguado, I. H. and Broers, B. and Boufassa, F. and Goldberg, D. J. and Zangerle, R. and Coutinho, R. A. and van den Hoek, A.},
	month = dec,
	year = {1999},
	pmid = {10597777},
	keywords = {Acquired Immunodeficiency Syndrome, Adult, Anti-HIV Agents, CD4 Lymphocyte Count, Cohort Studies, Female, HIV Infections, HIV Seropositivity, Humans, Male, Regression Analysis, Sex Characteristics, Time Factors},
	pages = {2361--2364},
}

@article{prins_gender_1999-1,
	title = {Do gender differences in {CD4} cell counts matter?},
	volume = {13},
	issn = {0269-9370},
	doi = {10.1097/00002030-199912030-00007},
	abstract = {OBJECTIVE: To examine the effect of gender on disease progression and whether gender differences in CD4 lymphocyte counts persisted for the entire course from HIV seroconversion until (death from) AIDS. METHODS: CD4 lymphocyte counts were modelled in 221 female and 443 male seroconverters following seroconversion, backwards from AIDS and backwards from death using regression analysis for repeated measurements. RESULTS: In the period before use of highly active antiretroviral therapy (HAART), progression to AIDS and to death were marginally slower in women than in men as assessed by proportional hazards analysis. Women seroconverted for HIV, developed AIDS and died at higher CD4 cell counts than men (women: 815, 146 and 44 x 10(6) cells/l, respectively; men: 727, 49 and 22 x 10(6) cells/l, respectively), although differences were only statistically significant at AIDS onset. Declines in CD4 lymphocyte counts were not significantly affected by gender and absolute differences between men and women were stable, with exception for the trajectory close to AIDS when the decline became steeper for men than women. CONCLUSION: These gender differences in CD4 lymphocyte counts suggest a delay of initiation of therapy in women compared with men (our model predicted that women reach the threshold of starting HAART at about 12 months later than men). If this delay unfavourably influences progression, treatment guidelines should be revised so that women can benefit equally from HAART.},
	language = {eng},
	number = {17},
	journal = {AIDS (London, England)},
	author = {Prins, M. and Robertson, J. R. and Brettle, R. P. and Aguado, I. H. and Broers, B. and Boufassa, F. and Goldberg, D. J. and Zangerle, R. and Coutinho, R. A. and van den Hoek, A.},
	month = dec,
	year = {1999},
	pmid = {10597777},
	keywords = {Acquired Immunodeficiency Syndrome, Adult, Anti-HIV Agents, CD4 Lymphocyte Count, Cohort Studies, Female, HIV Infections, HIV Seropositivity, Humans, Male, Regression Analysis, Sex Characteristics, Time Factors},
	pages = {2361--2364},
}

@article{ibrahim_missing_2009,
	title = {Missing data methods in longitudinal studies: a review},
	volume = {18},
	issn = {1133-0686},
	shorttitle = {Missing data methods in longitudinal studies},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3016756/},
	doi = {10.1007/s11749-009-0138-x},
	abstract = {Incomplete data are quite common in biomedical and other types of research, especially in longitudinal studies. During the last three decades, a vast amount of work has been done in the area. This has led, on the one hand, to a rich taxonomy of missing-data concepts, issues, and methods and, on the other hand, to a variety of data-analytic tools. Elements of taxonomy include: missing data patterns, mechanisms, and modeling frameworks; inferential paradigms; and sensitivity analysis frameworks. These are described in detail. A variety of concrete modeling devices is presented. To make matters concrete, two case studies are considered. The first one concerns quality of life among breast cancer patients, while the second one examines data from the Muscatine children’s obesity study.},
	number = {1},
	urldate = {2020-06-24},
	journal = {Test (Madrid, Spain)},
	author = {Ibrahim, Joseph G. and Molenberghs, Geert},
	month = may,
	year = {2009},
	pmid = {21218187},
	pmcid = {PMC3016756},
	pages = {1--43},
}

@article{yan_cd4_2016,
	title = {{CD4} measurement in a 54-year-old man with sustained {HIV}-1 viral suppression},
	volume = {188},
	issn = {0820-3946},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5047818/},
	doi = {10.1503/cmaj.151224},
	number = {14},
	urldate = {2020-06-24},
	journal = {CMAJ : Canadian Medical Association Journal},
	author = {Yan, Allan and Leis, Jerome A. and Gold, Wayne L.},
	month = oct,
	year = {2016},
	pmid = {27026669},
	pmcid = {PMC5047818},
	pages = {1028--1029},
}

@article{wright_ageing_2013,
	title = {Ageing \& long-term {CD4} cell count trends in {HIV}-positive patients with 5 years or more combination antiretroviral therapy experience},
	volume = {14},
	issn = {1464-2662},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3556360/},
	doi = {10.1111/j.1468-1293.2012.01053.x},
	abstract = {Background The aim of this analysis is to describe the long-term changes in CD4 cell counts beyond 5 years of combination antiretroviral therapy (cART). If natural ageing leads to a long-term decline in the immune system via low-grade chronic immune activation/inflammation, then one might expect to see a greater or earlier decline in CD4 counts in older HIV-positive patients with increasing duration of cART. Methods Retrospective and prospective data were examined from long-term virologically stable HIV-positive adults from the Australian HIV Observational Database. We estimated mean CD4 cell counts changes following the completion of 5 years of cART using linear mixed models. Results A total of 37,916 CD4 measurements were observed for 892 patients over a combined total of 9,753 patient years. Older patients ({\textbackslash}textgreater50 years) at cART initiation had estimated mean(95\% confidence interval) change in CD4 counts by Year-5 CD4 count strata ({\textbackslash}textless500, 501–750 and {\textbackslash}textgreater750 cells/μL) of 14(7 to 21), 3(−5 to 11) and −6(−17 to 4) cells/μL/year. Of the CD4 cell count rates of change estimated, none were indicative of long-term declines in CD4 cell counts. Conclusions Our results suggest that duration of cART and increasing age does not result in decreasing mean changes in CD4 cell counts for long-term virologically suppressed patients. Indicating that level of immune recovery achieved during the first 5 years of treatment are sustained through long-term cART.},
	number = {4},
	urldate = {2020-06-26},
	journal = {HIV medicine},
	author = {WRIGHT, ST and PETOUMENOS, K and BOYD, M and CARR, A and DOWNING, S and O’CONNOR, CC and GROTOWSKI, M and LAW, MG},
	month = apr,
	year = {2013},
	pmid = {23036045},
	pmcid = {PMC3556360},
	pages = {208--216},
}

@article{dolin_zidovudine_1995,
	title = {Zidovudine {Compared} {With} {Didanosine} in {Patients} {With} {Advanced} {HIV} {Type} {I} {Infection} and {Little} or {No} {Previous} {Experience} {With} {Zidovudine}},
	volume = {155},
	issn = {0003-9926},
	url = {https://jamanetwork.com/journals/jamainternalmedicine/fullarticle/620482},
	doi = {10.1001/archinte.1995.00430090111012},
	abstract = {{\textbackslash}textlessh3{\textbackslash}textgreaterBackground:{\textbackslash}textless/h3{\textbackslash}textgreater{\textbackslash}textlessp{\textbackslash}textgreaterWe conducted a trial to compare treatment with zidovudine or didanosine in patients with advanced human immunodeficiency virus type 1 (HIV-1) infection who had received little or no previous therapy with zidovudine.{\textbackslash}textless/p{\textbackslash}textgreater{\textbackslash}textlessh3{\textbackslash}textgreaterMethods:{\textbackslash}textless/h3{\textbackslash}textgreater{\textbackslash}textlessp{\textbackslash}textgreaterSix hundred seventeen patients with acquired immunodeficiency syndrome (AIDS), advanced AIDS-related complex (CD4 cell count, ≤0.30× 10⁹/L [300/μL]), or asymptomatic HIV (CD4 cell count, ≤0.20× 10⁹/L) received zidovudine, 500 mg/d of didanosine, or 750 mg/d of didanosine in a randomized, double-blind allocation, with cross-over to alternative medication after development of an end point or serious toxic effect. To be eligible, patients must have received either no or up to 16 weeks of zidovudine therapy before entry into the study. Primary end points were development of a new AIDS-defining event or death. Secondary clinical end points were new or recurrent AIDS-defining events, or death, and survival.{\textbackslash}textless/p{\textbackslash}textgreater{\textbackslash}textlessh3{\textbackslash}textgreaterResults:{\textbackslash}textless/h3{\textbackslash}textgreater{\textbackslash}textlessp{\textbackslash}textgreaterIn the study as a whole, there were no differences in the relative risks (RRs) of the development of end points between treatment groups. However, there was a strong interaction between the relative efficacies of zidovudine and didanosine and previous experience with zidovudine. Among 380 patients with no previous zidovudine therapy, zidovudine was more effective than 750 mg/d of didanosine (RR, 1.43; 90\% confidence interval [CI], 1.02 to 2.00), with a similar trend for zidovudine compared with 500 mg/d of didanosine (RR, 1.21; 90\% CI, 0.86 to 1.71). However, among 118 patients with more than 8 weeks but no more than 16 weeks of previous zidovudine therapy, 500 mg/d of didanosine was more effective than zidovudine (RR, 0.48; 90\% CI, 0.27 to 0.86); there was a similar trend for increased effectiveness of 750 mg/d of didanosine compared with zidovudine (RR, 0.61; 90\% CI, 0.36 to 1.03). Among 119 patients who had some but no more than 8 weeks of previous zidovudine therapy, there were no significant differences among the treatment arms. Similar findings were noted in the analysis of the two secondary clinical end points. No significant differences were found in efficacy between the groups receiving 500 and 750 mg/d of didanosine. The major toxic effect associated with zidovudine was hematopoietic (granulocytopenia) and that associated with didanosine was pancreatitis (dosage, 750 mg/d).{\textbackslash}textless/p{\textbackslash}textgreater{\textbackslash}textlessh3{\textbackslash}textgreaterConclusions:{\textbackslash}textless/h3{\textbackslash}textgreater{\textbackslash}textlessp{\textbackslash}textgreaterIn patients with advanced HIV disease, zidovudine appears to be more effective than didanosine as initial therapy; however, some patients with advanced HIV disease may benefit from a change to didanosine therapy after as little as 8 to 16 weeks of therapy with zidovudine.{\textbackslash}textless/p{\textbackslash}textgreater{\textbackslash}textlessp{\textbackslash}textgreater(Arch Intern Med. 1995;155:961-974){\textbackslash}textless/p{\textbackslash}textgreater},
	language = {en},
	number = {9},
	urldate = {2020-06-26},
	journal = {Archives of Internal Medicine},
	author = {Dolin, Raphael and Amato, David A. and Fischl, Margaret A. and Pettinelli, Carla and Beltangady, Mohan and Liou, Song-heng and Brown, Michael J. and Cross, Anne P. and Hirsch, Martin S. and Hardy, W. David and Mildvan, Donna and Blair, Donald C. and Powderly, William G. and Para, Michael F. and Fife, Kenneth H. and Steigbigel, Roy T. and Smaldone, Laurie and Crumpacker, Clyde S. and Cooley, Tim and Mitsuyaso, Ronald T. and John, Ric and Sanders, Carsandra and Reitman, Dinah and Hewitt, Ross and Reichman, Richard C. and Gelb, Lawrence D. and McGuire, M. L. and Jones, Marcella and Neidig, Judith L. and Zwickl, Beth and Hartman, Paula B. and Roarke, Mary Elizabeth and Burk, Ruth Ann and Fuhrer, Jack and Somogyi, Karen A. and Sepkowitz, Kent and Telzak, Edward E. and McAuliffe, Vincent J. and Valentine, Fred T. and Vasquez, Margarita and Merigan, Thomas C. and Katzenstein, David and Fessell, Jeffrey and Havlir, Diane V. and Richman, Douglas D. and Spector, Stephen A. and Kahn, James O. and Johnson, Linda and Coleman, Rebecca and Ho, Monto and McMahon, Deborah and Pazin, George and Antoniskis, Diana and McNamara, Bernard and Diamond, DeAnn and Collier, Ann C. and Paradise, Mary A. and Wald, Anna and DePaolis-Jones, Ann and Henry, Keith and Waskin, Hetty A. and Hyslop, Newton E. and Mushatt, David M. and Zachary, James A. and Soeiro, Roy and Harris, Carol and Zingman, Barry and Giordano, Michael F. and Sledz, Sandra and Murray, Henry W. and Carey, John T. and Davis, Traci L. and Bagby, Brenda and Kessler, Harold A. and Murphy, Robert I. and Hirschtick, Robert E. and Cheeseman, Sarah H. and Lai, Kwan Kew and Fairchild, Patrick G. and Ehmann, W. C. and Zurlo, J. J. and Millard, R. and Troiani, Luigi and Heggen, Aline A. and Horst, Charles M. van der and McKinley, George F. and Grieco, Michael H. and Kolatch, Brenda R. and Goldsmith, Jonathan C. and Gomperts, Edward D. and Woods, Lin M. and Grue, Louis and Mayjo, Kate and Becker, Rebecca L. and Jayaweera, D. T. and Rolfe, Lisa and Cole, Joanne and Jermano, John},
	month = may,
	year = {1995},
	pages = {961--974},
}

@incollection{li_cd4_2020,
	address = {Treasure Island (FL)},
	title = {{CD4} {Count}},
	copyright = {Copyright © 2020, StatPearls Publishing LLC.},
	url = {http://www.ncbi.nlm.nih.gov/books/NBK470231/},
	abstract = {CD4 count is a laboratory test that measures CD-4 T lymphocytes (T cells) via flow cytometry. This test is an important parameter in HIV management and is used to guide clinical treatment. The CD4 count is a reliable indicator of a patient’s immunologic status and is used to determine the necessity for initiation of prophylactic treatment against opportunistic infections.[1] CD4 T cells are human helper T cells that express cluster determinant 4 (CD4) molecules. The CD4 molecule is a member of the immunoglobulin family and primarily mediates adhesion to major histocompatibility complex molecules. CD4 T cells are selectively targeted and infected by HIV. HIV proliferates rapidly during acute infection leading to high levels of viremia and rapid impairment and death of CD4 T cells.[2][3][4][5]},
	language = {eng},
	urldate = {2020-06-26},
	booktitle = {{StatPearls}},
	publisher = {StatPearls Publishing},
	author = {Li, Raymund and Duffee, Doug and Gbadamosi-Akindele, Maryam F.},
	year = {2020},
	pmid = {29262137},
}

@article{adams_sex_2017,
	title = {Sex {Differences} in {Hospitalizations} with {Maintenance} {Hemodialysis}},
	volume = {28},
	issn = {1533-3450},
	doi = {10.1681/ASN.2016090986},
	abstract = {Hospitalization is a major source of morbidity among patients with ESRD undergoing maintenance hemodialysis and is a significant contributor to health care costs. To identify subgroups at the highest risk of hospitalization, we analyzed by sex, age, and race, adjusting for demographic and clinical characteristics, the hospitalization rates, and 30-day readmissions for 333,756 hospitalizations among 111,653 patients undergoing maintenance hemodialysis in facilities operated by a large dialysis organization in the United States (2007-2011). The overall hospitalization rate was 1.85 hospitalizations per person-year and was much higher among women than among men (2.08 versus 1.68 hospitalizations per year for women versus men, P{\textbackslash}textless0.001). Age group-specific hospitalization rates were consistently higher for women than for men of the same race, and the differences were greatest in younger age groups (for example, women aged 18-34 years and ≥75 years had 54\% [95\% confidence interval, 42\% to 67\%] and 14\% [95\% confidence interval, 11\% to 18\%] higher hospitalization rates, respectively, than did men of respective ages). Women also had substantially higher risk for 30-day readmission, with the largest differences at younger ages. Women had a significantly lower serum albumin level than men, and stratification by serum albumin level attenuated sex differences in the age group-specific hospitalization and 30-day readmission rates. These findings suggest that women undergoing maintenance hemodialysis have substantially higher risks for hospitalization and 30-day readmission than men. In this cohort, the sex differences were greatest in the younger age groups and were attenuated by accounting for differences in health status reflected by serum albumin level.},
	language = {eng},
	number = {9},
	journal = {Journal of the American Society of Nephrology: JASN},
	author = {Adams, Scott V. and Rivara, Matthew and Streja, Elani and Cheung, Alfred K. and Arah, Onyebuchi A. and Kalantar-Zadeh, Kamyar and Mehrotra, Rajnish},
	month = sep,
	year = {2017},
	pmid = {28432127},
	pmcid = {PMC5576928},
	keywords = {Adolescent, Adult, Age Factors, Aged, Chronic, Cohort Studies, Female, Health Status, Hospitalization, Humans, Kidney Failure, Male, Middle Aged, Patient Readmission, Renal Dialysis, Risk Factors, Serum Albumin, Sex Factors, United States, Young Adult, end stage kidney disease, hemodialysis, hospitalization},
	pages = {2721--2728},
}

@article{norris_racial_2008,
	title = {Racial {Differences} in {Mortality} and {End}-{Stage} {Renal} {Disease}},
	volume = {52},
	issn = {0272-6386},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2601720/},
	doi = {10.1053/j.ajkd.2008.06.004},
	number = {2},
	urldate = {2020-07-13},
	journal = {American journal of kidney diseases : the official journal of the National Kidney Foundation},
	author = {Norris, Keith and Mehrotra, Rajnish and Nissenson, Allen R.},
	month = aug,
	year = {2008},
	pmid = {18640483},
	pmcid = {PMC2601720},
	pages = {205--208},
}

@article{zhang_level_2019,
	title = {The {Level} of {Serum} {Albumin} {Is} {Associated} with {Renal} {Prognosis} in {Patients} with {Diabetic} {Nephropathy}},
	volume = {2019},
	issn = {2314-6745},
	url = {https://doi.org/10.1155/2019/7825804},
	doi = {10.1155/2019/7825804},
	abstract = {{\textbackslash}textlessitalic{\textbackslash}textgreaterObjective{\textbackslash}textless/italic{\textbackslash}textgreater. Although hypoalbuminemia is frequently found in most patients with diabetic nephropathy (DN), its relationship to the severity and progression of DN remains largely unknown. Our aim was to investigate the association between the serum albumin levels and clinicopathological features and renal outcomes in patients with type 2 diabetes mellitus (T2DM) and biopsy-proven DN. {\textbackslash}textlessitalic{\textbackslash}textgreaterMaterials and Methods{\textbackslash}textless/italic{\textbackslash}textgreater. A total of 188 patients with T2DM and biopsy-proven DN followed up for at least one year were enrolled. The patients were divided into four groups based on the albumin levels: normal group: \&\#x2265;35\&\#x2009;g/L ({\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreatern{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater87{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater); mild group: 30-35\&\#x2009;g/L ({\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreatern{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater34{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater); moderate group: 25-30\&\#x2009;g/L ({\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreatern{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater36{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater); and severe group: \&\#x003C;25\&\#x2009;g/L ({\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreatern{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater31{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater). The renal outcome was defined by progression to end-stage renal disease. The impact of the serum albumin level on renal survival was estimated using Cox regression analysis. {\textbackslash}textlessitalic{\textbackslash}textgreaterResults{\textbackslash}textless/italic{\textbackslash}textgreater. Among the cases, the serum albumin level had a significant correlation with proteinuria, renal function, and glomerular lesions. A multivariate Cox regression analysis indicated that the severity of hypoalbuminemia remained significantly associated with an adverse renal outcome, independent of clinical and histopathological features. In reference to the normal group, the risk of progression to ESRD increased such that the hazard ratio (HR) for the mild group was 2.09 (95\&\#x0025; CI, 0.67-6.56, {\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreaterp{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater0.205{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater), 6.20 (95\&\#x0025; CI, 1.95-19.76, {\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreaterp{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater0.002{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater) for the moderate group, and 7.37 (95\&\#x0025; CI, 1.24-43.83, {\textbackslash}textlessinline-formula{\textbackslash}textgreater{\textbackslash}textlessmml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"{\textbackslash}textgreater{\textbackslash}textlessmml:mi{\textbackslash}textgreaterp{\textbackslash}textless/mml:mi{\textbackslash}textgreater{\textbackslash}textlessmml:mo{\textbackslash}textgreater={\textbackslash}textless/mml:mo{\textbackslash}textgreater{\textbackslash}textlessmml:mn{\textbackslash}textgreater0.028{\textbackslash}textless/mml:mn{\textbackslash}textgreater{\textbackslash}textless/mml:math{\textbackslash}textgreater{\textbackslash}textless/inline-formula{\textbackslash}textgreater) for the severe group. {\textbackslash}textlessitalic{\textbackslash}textgreaterConclusions{\textbackslash}textless/italic{\textbackslash}textgreater. These findings suggested that hypoalbuminemia was associated with a poorer renal prognosis in patients with T2DM and DN.},
	journal = {Journal of Diabetes Research},
	author = {Zhang, Junlin and Zhang, Rui and Wang, Yiting and Li, Hanyu and Han, Qianqian and Wu, Yucheng and Wang, Tingli and Liu, Fang},
	editor = {Mastrocola, Raffaella},
	month = feb,
	year = {2019},
	pages = {7825804},
}

@article{chandrashekar_survival_2014,
	title = {Survival analysis of patients on maintenance hemodialysis},
	volume = {24},
	issn = {0971-4065},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4119332/},
	doi = {10.4103/0971-4065.132985},
	abstract = {Despite the continuous improvement of dialysis technology and pharmacological treatment, mortality rates for dialysis patients are still high. A 2-year prospective study was conducted at a tertiary care hospital to determine the factors influencing survival among patients on maintenance hemodialysis. 96 patients with end-stage renal disease surviving more than 3 months on hemodialysis (8-12 h/week) were studied. Follow-up was censored at the time of death or at the end of 2-year study period, whichever occurred first. Of the 96 patients studied (mean age 49.74 ± 14.55 years, 75\% male and 44.7\% diabetics), 19 died with an estimated mortality rate of 19.8\%. On an age-adjusted multivariate analysis, female gender and hypokalemia independently predicted mortality. In Cox analyses, patient survival was associated with delivered dialysis dose (single pool Kt/V, hazard ratio [HR] =0.01, P = 0.016), frequency of hemodialysis (HR = 3.81, P = 0.05) and serum albumin (HR = 0.24, P = 0.005). There was no significant difference between diabetes and non-diabetes in relation to death (Relative Risk = 1.109; 95\% CI = 0.49-2.48, P = 0.803). This study revealed that mortality among hemodialysis patients remained high, mostly due to sepsis and ischemic heart disease. Patient survival was better with higher dialysis dose, increased frequency of dialysis and adequate serum albumin level. Efforts at minimizing infectious complications, preventing cardiovascular events and improving nutrition should increase survival among hemodialysis patients.},
	number = {4},
	urldate = {2020-07-13},
	journal = {Indian Journal of Nephrology},
	author = {Chandrashekar, A. and Ramakrishnan, S. and Rangarajan, D.},
	year = {2014},
	pmid = {25097332},
	pmcid = {PMC4119332},
	pages = {206--213},
}

@misc{products_health_2019,
	title = {Health {Information}},
	url = {https://www.fda.gov/tobacco-products/public-health-education/health-information},
	abstract = {Tobacco use can cause health effects like cancer, heart disease and death. Tobacco can negatively impact people ranging from youth to pregnant women to seniors.},
	language = {en},
	urldate = {2020-07-13},
	author = {Products, Center for Tobacco},
	month = sep,
	year = {2019},
	note = {Publication Title: FDA},
}

@misc{noauthor_faststats_2020,
	title = {{FastStats}},
	url = {https://www.cdc.gov/nchs/fastats/leading-causes-of-death.htm},
	abstract = {FastStats is an official application from the Centers for Disease Control and Prevention’s (CDC) National Center for Health Statistics (NCHS) and puts access to topic-specific statistics at your fingertips.},
	language = {en-us},
	urldate = {2020-07-13},
	month = feb,
	year = {2020},
}

@article{prakash_interaction_2009,
	title = {Interaction of {Aging} and {CKD}},
	volume = {29},
	issn = {0270-9295},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2771919/},
	doi = {10.1016/j.semnephrol.2009.06.006},
	number = {5},
	urldate = {2020-07-13},
	journal = {Seminars in nephrology},
	author = {Prakash, Suma and O’Hare, Ann M.},
	month = sep,
	year = {2009},
	pmid = {19751895},
	pmcid = {PMC2771919},
	pages = {497--503},
}

@article{goldfarb-rumyantzev_duration_2005,
	title = {Duration of end-stage renal disease and kidney transplant outcome},
	volume = {20},
	issn = {0931-0509},
	doi = {10.1093/ndt/gfh541},
	abstract = {BACKGROUND: Patients nearing end-stage renal disease (ESRD) increasingly choose pre-emptive renal transplant (PRT) to avoid pre-transplant dialysis and to minimize ESRD. Compared with long-term dialysis, PRT has been shown to increase allograft survival. However, the merit of short-term dialysis is not well characterized, and it may be the better medical choice in some patients. The goal of the study was to characterize the relationship between the duration of dialysis vs allograft and patient survival. METHODS: We performed a retrospective nationwide cohort study of all kidney transplants (Tx) between January 1, 1990 and December 31, 1999, with a follow-up period through December 31, 2000. Participants were identified using the United States Renal Data System (USRDS), which tracks all ESRD cases in the nation including patients on dialysis and with kidney Tx. Patients with the history of more than one kidney Tx were excluded. Allograft survival and recipient survival were the primary outcomes of this study. Duration of ESRD as a continuous variable as well as divided into categories (14 days, 15-60 days, 61-180 days, 181-365 days, 1-2 years, 2-3 years, 3-5 years and {\textbackslash}textgreater5 years) was the primary risk factor of interest. Models were adjusted for multiple donor and recipient factors, including demographics and co-morbidities, as well as for Tx procedure characteristics. RESULTS: A total of 81,130 patient records were used for analysis (age 44.1+/-14.3 years, 61\% males, 24\% black, 29\% diabetic, pre-transplant ESRD duration 27.1+/-26.4 months, 26\% living donors). ESRD duration, as a continuous variable, is associated with a modest increase in the risk of graft failure over time [hazard ratio (HR) 1.02 per year of ESRD duration, P{\textbackslash}textless0.001]. When ESRD is studied as a categorical variable (duration of 0-14 days vs longer durations), the increased risk of allograft failure reached statistical significance only when the time on dialysis was {\textbackslash}textgreater or =181 days. The duration of ESRD was a significant risk for recipient death (HR 1.04 per year, P{\textbackslash}textless0.001); however, mortality risk reached statistical significance only when the patient had been on dialysis for {\textbackslash}textgreater or =1 year. CONCLUSIONS: This study of USRDS records suggests that a short ({\textbackslash}textless6 months) dialysis course has no detrimental effect on graft and patient survival, and should not be deferred if medically indicated.},
	language = {eng},
	number = {1},
	journal = {Nephrology, Dialysis, Transplantation: Official Publication of the European Dialysis and Transplant Association - European Renal Association},
	author = {Goldfarb-Rumyantzev, Alex and Hurdle, John F. and Scandling, John and Wang, Zhi and Baird, Bradley and Barenbaum, Lev and Cheung, Alfred K.},
	month = jan,
	year = {2005},
	pmid = {15546892},
	keywords = {Adult, Chronic, Cohort Studies, Female, Graft Rejection, Graft Survival, Humans, Kidney Failure, Kidney Function Tests, Kidney Transplantation, Male, Middle Aged, Probability, Prognosis, Proportional Hazards Models, Registries, Renal Dialysis, Retrospective Studies, Risk Assessment, Severity of Illness Index, Survival Rate, Time Factors, United States},
	pages = {167--175},
}

@misc{noauthor_kidney_nodate,
	title = {Kidney {Disease} {Statistics} for the {United} {States} {\textbackslash}textbar {NIDDK}},
	url = {https://www.niddk.nih.gov/health-information/health-statistics/kidney-disease},
	abstract = {Provides statistics about kidney disease, including prevalence and survival rates.},
	language = {en-US},
	urldate = {2020-07-14},
	note = {Publication Title: National Institute of Diabetes and Digestive and Kidney Diseases},
}

@article{fan_comparison_2013,
	title = {A {Comparison} of {Bias}-{Corrected} {Covariance} {Estimators} for {Generalized} {Estimating} {Equations}},
	volume = {23},
	issn = {1054-3406},
	url = {https://doi.org/10.1080/10543406.2013.813521},
	doi = {10.1080/10543406.2013.813521},
	abstract = {Although asymptotically the sandwich covariance estimator is consistent and robust with respect to the selection of the working correlation matrix, when the sample size is small, its bias may not be negligible. This article compares the small sample corrections for the sandwich covariance estimator as well as the inferential procedures proposed by Mancl and DeRouen (2001), Kauermann and Carroll (2001), Fay and Graubard (2001), and Fan et al. (2012). Simulation studies show that when using a maximum likelihood method to estimate the covariance parameters and using the between-within method for the denominator degrees of freedom when making inference, the Kauermann and Carroll method is preferred in the investigated balanced logistic regression and the Mancl and DeRouen and Fan et al. methods are preferred in the investigated proportional odds model. A collagen-induced arthritis study is employed to demonstrate the application of the methods.},
	number = {5},
	urldate = {2020-07-15},
	journal = {Journal of Biopharmaceutical Statistics},
	author = {Fan, Chunpeng and Zhang, Donghui and Zhang, Cun-Hui},
	month = sep,
	year = {2013},
	pmid = {23957522},
	keywords = {Empirical covariance estimator, GEE, Sandwich covariance estimator},
	pages = {1172--1187},
}

@article{kauermann_note_2001,
	title = {A {Note} on the {Efficiency} of {Sandwich} {Covariance} {Matrix} {Estimation}},
	volume = {96},
	issn = {0162-1459},
	url = {https://www.tandfonline.com/doi/10.1198/016214501753382309},
	doi = {10.1198/016214501753382309},
	abstract = {The sandwich estimator, also known as robust covariance matrix estimator, heteroscedasticity-consistent covariance matrix estimate, or empirical covariance matrix estimator, has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations. Its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified. Surprisingly though, there has been little discussion of properties of the sandwich method other than consistency. We investigate the sandwich estimator in quasi-likelihood models asymptotically, and in the linear case analytically. We show that under certain circumstances when the quasi-likelihood model is correct, the sandwich estimate is often far more variable than the usual parametric variance estimate. The increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the parametric model fails or when there is heteroscedasticity. We show that the additional variability directly affects the coverage probability of confidence intervals constructed from sandwich variance estimates. In fact, the use of sandwich variance estimates combined with t-distribution quantiles gives confidence intervals with coverage probability falling below the nominal value. We propose an adjustment to compensate for this fact.},
	number = {456},
	urldate = {2020-07-15},
	journal = {Journal of the American Statistical Association},
	author = {Kauermann, Göran and Carroll, Raymond J},
	month = dec,
	year = {2001},
	pages = {1387--1396},
}

@misc{prevention_cdc_2020,
	title = {{CDC} - {Malaria} - {FAQs}},
	url = {https://www.cdc.gov/malaria/about/faqs.html},
	abstract = {Frequently asked questions about malaria, transmission, symptoms, diagnosis, treatment, complications with pregnancy, prevention, eradication and endemic areas.},
	language = {en-us},
	urldate = {2020-07-15},
	author = {Prevention, CDC-Centers for Disease Control and},
	month = mar,
	year = {2020},
}

@article{cunha_osler_2004,
	series = {Historical {Aspects} of {Infectious} {Diseases}, {Part} {I}},
	title = {Osler on typhoid fever: differentiating typhoid from typhus and malaria},
	volume = {18},
	issn = {0891-5520},
	shorttitle = {Osler on typhoid fever},
	url = {http://www.sciencedirect.com/science/article/pii/S0891552003000941},
	doi = {10.1016/S0891-5520(03)00094-1},
	language = {en},
	number = {1},
	urldate = {2020-07-15},
	journal = {Infectious Disease Clinics of North America},
	author = {Cunha, Burke A.},
	month = mar,
	year = {2004},
	pages = {111--125},
}

@article{mancl_covariance_2001,
	title = {A {Covariance} {Estimator} for {GEE} with {Improved} {Small}-{Sample} {Properties}},
	volume = {57},
	issn = {1541-0420},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/j.0006-341X.2001.00126.x},
	doi = {10.1111/j.0006-341X.2001.00126.x},
	abstract = {Summary. In this paper, we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations (GEE). Hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small. Resampling methods, such as the jackknife and bootstrap, have been suggested for covariance estimation when the number of clusters is small. A drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near-zero cell counts caused by resampling. We propose a bias-corrected covariance estimator that avoids this problem. In a small simulation study, we compare the bias-corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10–40 subjects with equal and unequal cluster sizes of 16–64 observations. The bias-corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal, whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2–3 times the nominal level. The methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease.},
	language = {en},
	number = {1},
	urldate = {2020-07-16},
	journal = {Biometrics},
	author = {Mancl, Lloyd A. and DeRouen, Timothy A.},
	year = {2001},
	keywords = {Correlated binary data, Logistic regression, Periodontal disease, Simulation study},
	pages = {126--134},
}

@article{lunardon_comment_2017,
	title = {Comment on ‘{Small} sample {GEE} estimation of regression parameters for longitudinal data’},
	volume = {36},
	copyright = {Copyright © 2017 John Wiley \& Sons, Ltd.},
	issn = {1097-0258},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.7366},
	doi = {10.1002/sim.7366},
	abstract = {In longitudinal studies, the generalized estimating equation (GEE) estimator of the parameters of a marginal model is known to be consistent even if the working intra-subject covariance matrix is incorrectly specified. Recently, a small sample correction for the bias of the GEE estimator has been proposed. We show that this correction formula relies on the correct specification of the working intra-subject covariance matrix. We provide a revised formula that is valid under misspecification and develop the R package ‘BCgee’ to ease the practical use of the formula. Copyright © 2017 John Wiley \& Sons, Ltd.},
	language = {en},
	number = {22},
	urldate = {2020-07-16},
	journal = {Statistics in Medicine},
	author = {Lunardon, N. and Scharfstein, D.},
	year = {2017},
	pages = {3596--3600},
}

@article{yeshiwondim_spatial_2009,
	title = {Spatial analysis of malaria incidence at the village level in areas with unstable transmission in {Ethiopia}},
	volume = {8},
	issn = {1476-072X},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2646707/},
	doi = {10.1186/1476-072X-8-5},
	abstract = {Background Malaria is the leading cause of morbidity and mortality in Ethiopia, accounting for over five million cases and thousands of deaths annually. The risks of morbidity and mortality associated with malaria are characterized by spatial and temporal variation across the country. This study examines the spatial and temporal patterns of malaria transmission at the local level and implements a risk mapping tool to aid in monitoring and disease control activities. Methods In this study, we examine the global and local patterns of malaria distribution in 543 villages in East Shoa, central Ethiopia using individual-level morbidity data collected from six laboratory and treatment centers between September 2002 and August 2006. Results Statistical analysis of malaria incidence by sex, age, and village through time reveal the presence of significant spatio-temporal variations. Poisson regression analysis shows a decrease in malaria incidence with increasing age. A significant difference in the malaria incidence density ratio (IDRs) is detected in males but not in females. A significant decrease in the malaria IDRs with increasing age is captured by a quadratic model. Local spatial statistics reveals clustering or hot spots within a 5 and 10 km distance of most villages in the study area. In addition, there are temporal variations in malaria incidence. Conclusion Malaria incidence varies according to gender and age, with males age 5 and above showing a statistically higher incidence. Significant local clustering of malaria incidence occurs between pairs of villages within 1–10 km distance lags. Malaria incidence was higher in 2002–2003 than in other periods of observation. Malaria hot spots are displayed as risk maps that are useful for monitoring and spatial targeting of prevention and control measures against the disease.},
	urldate = {2020-07-17},
	journal = {International Journal of Health Geographics},
	author = {Yeshiwondim, Asnakew K and Gopal, Sucharita and Hailemariam, Afework T and Dengela, Dereje O and Patel, Hrishikesh P},
	month = jan,
	year = {2009},
	pmid = {19171051},
	pmcid = {PMC2646707},
	pages = {5},
}

@misc{noauthor_many_nodate,
	title = {Many {Factors} {Affect} {MPG}},
	url = {https://www.fueleconomy.gov/feg/factors.shtml},
	urldate = {2020-10-19},
}

@misc{noauthor_world_nodate,
	title = {World malaria report 2019},
	url = {https://www.who.int/publications/i/item/world-malaria-report-2019},
	urldate = {2020-07-17},
}

@misc{canada_factors_2018,
	title = {Factors that affect fuel efficiency},
	url = {https://www.nrcan.gc.ca/energy/efficiency/energy-efficiency-transportation-and-alternative-fuels/choosing-right-vehicle/tips-buying-fuel-efficient-vehicle/factors-affect-fuel-efficiency/21012},
	language = {eng},
	urldate = {2020-10-19},
	author = {Canada, Natural Resources},
	month = apr,
	year = {2018},
}

@misc{noauthor_save_nodate,
	title = {Save money and fuel by driving efficiently.},
	url = {//www.fueleconomy.gov/feg/driveHabits.jsp},
	abstract = {Our fuel-saving tips can help you save both fuel and money!},
	language = {en},
	urldate = {2020-10-19},
}

@article{elmqvist_rolling_2008,
	title = {Rolling the {Dice}: {Multidimensional} {Visual} {Exploration} using {Scatterplot} {Matrix} {Navigation}},
	volume = {14},
	issn = {1941-0506},
	shorttitle = {Rolling the {Dice}},
	doi = {10.1109/TVCG.2008.153},
	abstract = {Scatterplots remain one of the most popular and widely-used visual representations for multidimensional data due to their simplicity, familiarity and visual clarity, even if they lack some of the flexibility and visual expressiveness of newer multidimensional visualization techniques. This paper presents new interactive methods to explore multidimensional data using scatterplots. This exploration is performed using a matrix of scatterplots that gives an overview of the possible configurations, thumbnails of the scatterplots, and support for interactive navigation in the multidimensional space. Transitions between scatterplots are performed as animated rotations in 3D space, somewhat akin to rolling dice. Users can iteratively build queries using bounding volumes in the dataset, sculpting the query from different viewpoints to become more and more refined. Furthermore, the dimensions in the navigation space can be reordered, manually or automatically, to highlight salient correlations and differences among them. An example scenario presents the interaction techniques supporting smooth and effortless visual exploration of multidimensional datasets.},
	number = {6},
	journal = {IEEE Transactions on Visualization and Computer Graphics},
	author = {Elmqvist, Niklas and Dragicevic, Pierre and Fekete, Jean-Daniel},
	month = nov,
	year = {2008},
	keywords = {Animation, Data visualization, Graphics, Index Terms—, Light scattering, Multidimensional systems, Navigation, Optical polarization, Rendering (computer graphics), Visual analytics, Vocabulary, animated rotations, bounding volumes, computer animation, data visualisation, interaction, multidimensional visual exploration, multivariate data, navigation, scatterplot matrix navigation, visual analytics, visual exploration, visual queries},
	pages = {1539--1148},
}

@article{li_judging_2010,
	title = {Judging {Correlation} from {Scatterplots} and {Parallel} {Coordinate} {Plots}},
	volume = {9},
	issn = {1473-8716},
	url = {https://doi.org/10.1057/ivs.2008.13},
	doi = {10.1057/ivs.2008.13},
	abstract = {Scatterplots and parallel coordinate plots (PCPs) that can both be used to assess correlation visually. In this paper, we compare these two visualization methods in a controlled user experiment. More specifically, 25 participants were asked to report observed correlation as a function of the sample correlation under varying conditions of visualization method, sample size and observation time. A statistical model is proposed to describe the correlation judgment process. The accuracy and the bias in the judgments in different conditions are established by interpreting the parameters in this model. A discriminability index is proposed to characterize the performance accuracy in each experimental condition. Moreover, a statistical test is applied to derive whether or not the human sensation scale differs from a theoretically optimal (that is, unbiased) judgment scale. Based on these analyses, we conclude that users can reliably distinguish twice as many different correlation levels when using scatterplots as when using PCPs. We also find that there is a bias towards reporting negative correlations when using PCPs. Therefore, we conclude that scatterplots are more effective than parallel plots in supporting visual correlation analysis.},
	number = {1},
	urldate = {2020-10-19},
	journal = {Information Visualization},
	author = {Li, Jing and Martens, Jean-Bernard and van Wijk, Jarke J},
	month = jan,
	year = {2010},
	pages = {13--30},
}

@article{thaker_communicating_2016,
	title = {Communicating {Value} in {Health} {Care} {Using} {Radar} {Charts}: {A} {Case} {Study} of {Prostate} {Cancer}},
	volume = {12},
	issn = {1554-7477},
	shorttitle = {Communicating {Value} in {Health} {Care} {Using} {Radar} {Charts}},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5508207/},
	doi = {10.1200/JOP.2016.011320},
	abstract = {Purpose: The transformation from volume to value will require communication of outcomes and costs of therapies; however, outcomes are usually nonstandardized, and cost of therapy differs among stakeholders. We developed a standardized value framework by using radar charts to visualize and communicate a wide range of patient outcomes and cost for three forms of prostate cancer treatment. Materials and Methods: We retrospectively reviewed data from men with low-risk prostate cancer who were treated with low-dose rate brachytherapy (LDR-BT), proton beam therapy, or robotic-assisted prostatectomy. Patient-reported outcomes comprised the Expanded Prostate Cancer Index Composite-50 domains for sexual function, urinary incontinence and/or bother, bowel bother, and vitality 12 months after treatment. Costs were measured by time-driven activity-based costing for the first 12 months of the care cycle. Outcome and cost data were plotted on a single radar chart for each treatment modality. Results: Outcome and cost data from patients who were treated with robotic-assisted prostatectomy (n = 381), proton beam therapy (n = 165), and LDR-BT (n = 238) were incorporated into the radar chart. LDR-BT seemed to deliver the highest overall value of the three treatment modalities; however, incorporation of patient preferences regarding outcomes may allow other modalities to be considered high-value treatment options. Conclusion: Standardization and visualization of outcome and cost metrics may allow more comprehensive and collaborative discussions about the value of health care services. Communicating the value framework by using radar charts may be an effective method to present total value and the value of all outcomes and costs in a manner that is accessible to all stakeholders. Variations in plotting of costs and outcomes will require future focus group initiatives.},
	number = {9},
	urldate = {2020-10-19},
	journal = {Journal of Oncology Practice},
	author = {Thaker, Nikhil G. and Ali, Tariq N. and Porter, Michael E. and Feeley, Thomas W. and Kaplan, Robert S. and Frank, Steven J.},
	month = sep,
	year = {2016},
	pmid = {27577622},
	pmcid = {PMC5508207},
	pages = {813--820},
}

@article{fisher_use_1936,
	title = {The {Use} of {Multiple} {Measurements} in {Taxonomic} {Problems}},
	volume = {7},
	copyright = {1936 Blackwell Publishing Ltd/University College London},
	issn = {2050-1439},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-1809.1936.tb02137.x},
	doi = {10.1111/j.1469-1809.1936.tb02137.x},
	abstract = {The articles published by the Annals of Eugenics (1925–1954) have been made available online as an historical archive intended for scholarly use. The work of eugenicists was often pervaded by prejudice against racial, ethnic and disabled groups. The online publication of this material for scholarly research purposes is not an endorsement of those views nor a promotion of eugenics in any way.},
	language = {en},
	number = {2},
	urldate = {2020-10-20},
	journal = {Annals of Eugenics},
	author = {Fisher, R. A.},
	year = {1936},
	pages = {179--188},
}

@article{yao_yes_2018,
	title = {Yes, but {Did} {It} {Work}?: {Evaluating} {Variational} {Inference}},
	shorttitle = {Yes, but {Did} {It} {Work}?},
	url = {http://arxiv.org/abs/1802.02538},
	abstract = {While it's always possible to compute a variational approximation to a posterior distribution, it can be difficult to discover problems with this approximation. We propose two diagnostic algorithms to alleviate this problem. The Pareto-smoothed importance sampling (PSIS) diagnostic gives a goodness of fit measurement for joint distributions, while simultaneously improving the error in the estimate. The variational simulation-based calibration (VSBC) assesses the average performance of point estimates.},
	urldate = {2020-11-14},
	journal = {arXiv:1802.02538 [stat]},
	author = {Yao, Yuling and Vehtari, Aki and Simpson, Daniel and Gelman, Andrew},
	month = jul,
	year = {2018},
	keywords = {Statistics - Computation, Statistics - Machine Learning},
}

@article{klasnja_microrandomized_2015,
	title = {Microrandomized trials: {An} experimental design for developing just-in-time adaptive interventions},
	volume = {34S},
	issn = {1930-7810},
	shorttitle = {Microrandomized trials},
	doi = {10.1037/hea0000305},
	abstract = {OBJECTIVE: This article presents an experimental design, the microrandomized trial, developed to support optimization of just-in-time adaptive interventions (JITAIs). JITAIs are mHealth technologies that aim to deliver the right intervention components at the right times and locations to optimally support individuals' health behaviors. Microrandomized trials offer a way to optimize such interventions by enabling modeling of causal effects and time-varying effect moderation for individual intervention components within a JITAI. METHOD: The article describes the microrandomized trial design, enumerates research questions that this experimental design can help answer, and provides an overview of the data analyses that can be used to assess the causal effects of studied intervention components and investigate time-varying moderation of those effects. RESULTS: Microrandomized trials enable causal modeling of proximal effects of the randomized intervention components and assessment of time-varying moderation of those effects. CONCLUSION: Microrandomized trials can help researchers understand whether their interventions are having intended effects, when and for whom they are effective, and what factors moderate the interventions' effects, enabling creation of more effective JITAIs.},
	language = {eng},
	journal = {Health Psychology: Official Journal of the Division of Health Psychology, American Psychological Association},
	author = {Klasnja, Predrag and Hekler, Eric B. and Shiffman, Saul and Boruvka, Audrey and Almirall, Daniel and Tewari, Ambuj and Murphy, Susan A.},
	month = dec,
	year = {2015},
	pmid = {26651463},
	pmcid = {PMC4732571},
	keywords = {Adaptation, Early Medical Intervention, Health Behavior, Humans, Psychological, Randomized Controlled Trials as Topic, Research Design, Telemedicine},
	pages = {1220--1228},
}

@article{gershman_tutorial_2011,
	title = {A {Tutorial} on {Bayesian} {Nonparametric} {Models}},
	url = {http://arxiv.org/abs/1106.2697},
	abstract = {A key problem in statistical modeling is model selection, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number ofclusters in mixture models or the number of factors in factor analysis. In this tutorial we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduction to Bayesian nonparametric methods and contains several examples of their application.},
	urldate = {2021-03-08},
	journal = {arXiv:1106.2697 [stat]},
	author = {Gershman, Samuel J. and Blei, David M.},
	month = aug,
	year = {2011},
	keywords = {Statistics - Machine Learning, Statistics - Methodology},
}

@article{zhang_very_nodate,
	title = {A {Very} {Gentle} {Note} on the {Construction} of {Dirichlet} {Process}},
	language = {en},
	author = {Zhang, Xinhua},
	pages = {15},
}

@article{shotwell_profdpm_2013,
	title = {profdpm: {An} {R} {Package} for {MAP} {Estimation} in a {Class} of {Conjugate} {Product} {Partition} {Models}},
	volume = {53},
	copyright = {Copyright (c) 2011 Matthew S. Shotwell},
	issn = {1548-7660},
	shorttitle = {profdpm},
	url = {https://www.jstatsoft.org/index.php/jss/article/view/v053i08},
	doi = {10.18637/jss.v053.i08},
	language = {en},
	number = {1},
	urldate = {2021-03-08},
	journal = {Journal of Statistical Software},
	author = {Shotwell, Matthew S.},
	month = may,
	year = {2013},
	pages = {1--18},
}

@article{pereira_concentration_2018,
	title = {Concentration in the {Generalized} {Chinese} {Restaurant} {Process}},
	url = {http://arxiv.org/abs/1806.09656},
	abstract = {The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers \{\vphantom{\}}{\textbackslash}textbackslash\{1,{\textbackslash}textbackslashdots,n{\textbackslash}textbackslash\}{\textbackslash}. This process is related to the Ewens sampling model in Genetics and to Bayesian nonparametric methods such as topic models. In this paper, we study the GCRP in a regime where the number of parts grows like {\textbackslash}nˆ{\textbackslash}textbackslashalpha{\textbackslash} with \{\vphantom{\}}{\textbackslash}textbackslashalpha{\textbackslash}textgreater0{\textbackslash}. We prove a non-asymptotic concentration result for the number of parts of size {\textbackslash}k=o(nˆ\{{\textbackslash}textbackslashalpha/(2{\textbackslash}textbackslashalpha+4)\}/({\textbackslash}textbackslashlog n)ˆ\{1/(2+{\textbackslash}textbackslashalpha)\}){\textbackslash}. In particular, we show that these random variables concentrate around {\textbackslash}c\_\{k\}{\textbackslash}textbackslash,V\_*{\textbackslash}textbackslash,nˆ{\textbackslash}textbackslashalpha{\textbackslash} where {\textbackslash}V\_*{\textbackslash}textbackslash,nˆ{\textbackslash}textbackslashalpha{\textbackslash} is the asymptotic number of parts and {\textbackslash}c\_k{\textbackslash}textbackslashapprox kˆ\{-(1+{\textbackslash}textbackslashalpha)\}{\textbackslash} is a positive value depending on {\textbackslash}k{\textbackslash}. We also obtain finite-{\textbackslash}n{\textbackslash} bounds for the total number of parts. Our theorems complement asymptotic statements by Pitman and more recent results on large and moderate deviations by Favaro, Feng and Gao.},
	urldate = {2021-03-08},
	journal = {arXiv:1806.09656 [math]},
	author = {Pereira, Alan and Oliveira, Roberto I. and Ribeiro, Rodrigo},
	month = jun,
	year = {2018},
	keywords = {Mathematics - Probability},
}

@misc{noauthor_mstm_aoaspdf_nodate,
	title = {{MSTM}\_AOAS.pdf},
	url = {https://drive.google.com/file/d/0B3VBas8nQr8wRlF5MFVkcG1uaUk/view?usp=embed_facebook},
	urldate = {2021-03-22},
	note = {Publication Title: Google Docs},
}

@misc{noauthor_201819_nodate,
	title = {2018/19 {NBA} {Player} {Salaries} {\textbackslash}textbar {HoopsHype}},
	url = {https://hoopshype.com/salaries/players/2018-2019/},
	urldate = {2021-03-22},
}

@incollection{dahl_model-based_2006,
	address = {Cambridge},
	title = {Model-{Based} {Clustering} for {Expression} {Data} via a {Dirichlet} {Process} {Mixture} {Model}},
	isbn = {978-0-511-58458-9},
	url = {https://www.cambridge.org/core/product/identifier/CBO9780511584589A070/type/book_part},
	abstract = {This chapter describes a clustering procedure for microarray expression data based on a well-deﬁned statistical model, speciﬁcally, a conjugate Dirichlet process mixture model. The clustering algorithm groups genes whose latent variables governing expression are equal, that is, genes belonging to the same mixture component. The model is ﬁt with Markov chain Monte Carlo and the computational burden is eased by exploiting conjugacy. This chapter introduces a method to get a point estimate of the true clustering based on least-squares distances from the posterior probability that two genes are clustered. Unlike ad hoc clustering methods, the model provides measures of uncertainty about the clustering. Further, the model automatically estimates the number of clusters and quantiﬁes uncertainty about this important parameter. The method is compared to other clustering methods in a simulation study. Finally, the method is demonstrated with actual microarray data.},
	language = {en},
	urldate = {2021-03-22},
	booktitle = {Bayesian {Inference} for {Gene} {Expression} and {Proteomics}},
	publisher = {Cambridge University Press},
	author = {Dahl, David B. and Vannucci, Marina},
	editor = {Do, Kim-Anh and Muller, Peter},
	year = {2006},
	doi = {10.1017/CBO9780511584589.011},
	pages = {201--218},
}

@misc{noauthor_2018-19_nodate,
	title = {2018-19 {NBA} {Season} {Summary}},
	url = {https://www.basketball-reference.com/leagues/NBA_2019.html},
	abstract = {Statistics, leaders, and more for the 2018-19 NBA season},
	language = {en},
	urldate = {2021-03-22},
	note = {Publication Title: Basketball-Reference.com},
}

@article{noauthor_notitle_nodate,
}

@article{casella_introduction_1985,
	title = {An {Introduction} to {Empirical} {Bayes} {Data} {Analysis}},
	volume = {39},
	issn = {0003-1305},
	url = {https://www.jstor.org/stable/2682801},
	doi = {10.2307/2682801},
	abstract = {Empirical Bayes methods have been shown to be powerful data-analysis tools in recent years. The empirical Bayes model is much richer than either the classical or the ordinary Bayes model and often provides superior estimates of parameters. An introduction to some empirical Bayes methods is given, and these methods are illustrated with two examples.},
	number = {2},
	urldate = {2021-07-16},
	journal = {The American Statistician},
	author = {Casella, George},
	year = {1985},
	pages = {83--87},
}

@article{zhang_study_2021,
	title = {A {Study} on the {Association} between {Maternal} {Childhood} {Trauma} {Exposure} and {Placental}-fetal {Stress} {Physiology} during {Pregnancy}},
	url = {http://arxiv.org/abs/2101.12451},
	abstract = {It has been found that the effect of childhood trauma (CT) exposure may pass on to the next generation. Scientists have hypothesized that the association between CT exposure and placental-fetal stress physiology is the mechanism. A study was conducted to examine the hypothesis. To examine the association between CT exposure and placental corticotrophin-releasing hormone (pCRH), linear mixed effect model and hierarchical Bayesian linear model were constructed. In Bayesian inference, by providing conditionally conjugate priors, Gibbs sampler was used to draw MCMC samples. Piecewise linear mixed effect model was conducted in order to adjust to the dramatic change of pCRH at around week 20 into pregnancy. Pearson residual, QQ, ACF and trace plots were used to justify the model adequacy. Likelihood ratio test and DIC were utilized to model selection. The association between CT exposure and pCRH during pregnancy is obvious. The effect of CT exposure on pCRH varies dramatically over gestational age. Women with one childhood trauma would experience 11.9\% higher in pCRH towards the end of pregnancy than those without childhood trauma. The increase rate of pCRH after week 20 is almost four-fold larger than that before week 20. Frequentist and Bayesian inference produce similar results. The findings support the hypothesis that the effect of CT exposure on pCRH over GA exists. The effect changes dramatically at around week 20 into pregnancy.},
	urldate = {2022-04-14},
	journal = {arXiv:2101.12451 [stat]},
	author = {Zhang, Eileen},
	month = jan,
	year = {2021},
	keywords = {Statistics - Applications},
}

@article{zhang_study_2021-1,
	title = {A {Study} on the {Association} between {Maternal} {Childhood} {Trauma} {Exposure} and {Placental}-fetal {Stress} {Physiology} during {Pregnancy}},
	url = {https://arxiv.org/abs/2101.12451v1},
	doi = {10.48550/arXiv.2101.12451},
	abstract = {It has been found that the effect of childhood trauma (CT) exposure may pass on to the next generation. Scientists have hypothesized that the association between CT exposure and placental-fetal stress physiology is the mechanism. A study was conducted to examine the hypothesis. To examine the association between CT exposure and placental corticotrophin-releasing hormone (pCRH), linear mixed effect model and hierarchical Bayesian linear model were constructed. In Bayesian inference, by providing conditionally conjugate priors, Gibbs sampler was used to draw MCMC samples. Piecewise linear mixed effect model was conducted in order to adjust to the dramatic change of pCRH at around week 20 into pregnancy. Pearson residual, QQ, ACF and trace plots were used to justify the model adequacy. Likelihood ratio test and DIC were utilized to model selection. The association between CT exposure and pCRH during pregnancy is obvious. The effect of CT exposure on pCRH varies dramatically over gestational age. Women with one childhood trauma would experience 11.9\% higher in pCRH towards the end of pregnancy than those without childhood trauma. The increase rate of pCRH after week 20 is almost four-fold larger than that before week 20. Frequentist and Bayesian inference produce similar results. The findings support the hypothesis that the effect of CT exposure on pCRH over GA exists. The effect changes dramatically at around week 20 into pregnancy.},
	language = {en},
	urldate = {2022-04-14},
	author = {Zhang, Eileen},
	month = jan,
	year = {2021},
}

@article{moog_maternal_2016,
	series = {Mechanisms of {Resilience} to {Stress} {Effects}},
	title = {Maternal {Exposure} to {Childhood} {Trauma} {Is} {Associated} {During} {Pregnancy} {With} {Placental}-{Fetal} {Stress} {Physiology}},
	volume = {79},
	issn = {0006-3223},
	url = {https://www.sciencedirect.com/science/article/pii/S0006322315007313},
	doi = {10.1016/j.biopsych.2015.08.032},
	abstract = {Background The effects of exposure to childhood trauma (CT) may be transmitted across generations; however, the time period(s) and mechanism(s) have yet to be clarified. We address the hypothesis that intergenerational transmission may begin during intrauterine life via the effect of maternal CT exposure on placental-fetal stress physiology, specifically placental corticotropin-releasing hormone (pCRH). Methods The study was conducted in a sociodemographically diverse cohort of 295 pregnant women. CT exposure was assessed using the Childhood Trauma Questionnaire. Placental CRH concentrations were quantified in maternal blood collected serially over the course of gestation. Linear mixed effects and Bayesian piece-wise linear models were employed to test hypothesized relationships. Results Maternal CT exposure (CT+) was significantly associated with pCRH production. Compared with nonexposed women, CT+ was associated with an almost 25\% increase in pCRH toward the end of gestation, and the pCRH trajectory of CT+ women exhibited an approximately twofold steeper increase after the pCRH inflection point at 19 weeks gestation. Conclusions To the best of our knowledge, this finding represents the first report linking maternal CT exposure with placental-fetal stress physiology, thus identifying a potential novel biological pathway of intergenerational transmission that may operate as early as during intrauterine life.},
	language = {en},
	number = {10},
	urldate = {2022-04-12},
	journal = {Biological Psychiatry},
	author = {Moog, Nora K. and Buss, Claudia and Entringer, Sonja and Shahbaba, Babak and Gillen, Daniel L. and Hobel, Calvin J. and Wadhwa, Pathik D.},
	month = may,
	year = {2016},
	keywords = {Childhood trauma, Developmental programming, Intergenerational transmission, Placental CRH, Preconceptional stress, Pregnancy},
	pages = {831--839},
}

@misc{noauthor_chemical_nodate,
	title = {Chemical {Commonly} {Found} in {Consumer} {Products} {May} {Disrupt} a {Hormone} {Needed} for {Healthy} {Pregnancy}},
	url = {https://www.rutgers.edu/news/chemical-commonly-found-consumer-products-may-disrupt-hormone-needed-healthy-pregnancy},
	language = {en},
	urldate = {2022-04-12},
}

@article{steine_maternal_2020,
	title = {Maternal exposure to childhood traumatic events, but not multi-domain psychosocial stressors, predict placental corticotrophin releasing hormone across pregnancy},
	volume = {266},
	issn = {0277-9536},
	url = {https://www.sciencedirect.com/science/article/pii/S0277953620306808},
	doi = {10.1016/j.socscimed.2020.113461},
	abstract = {Maternal psychosocial stress increases the risk of adverse birth and postnatal outcomes for the mother and child, but the role of maternal exposure to childhood traumatic events (CTE) and multi-domain psychosocial stressors for the level and rise of placental Corticotrophin-Releasing Hormone (pCRH) across pregnancy has been understudied. In a sociodemographically and racially diverse sample of 1303 women (64\% Black, 36\% White/others) with low-medical risk pregnancies at enrollment from Shelby County, Tennessee, USA, blood samples were drawn twice, corresponding roughly to second and third trimester, and extracted prior to conducting radioimmune assays for pCRH. Mothers reported CTE (physical abuse, sexual abuse, or family violence, in childhood), adulthood traumatic events, and interpersonal violence during pregnancy. Neighborhood crime/deprivation was derived using geospatially-linked objective databases. General linear and mixed models tested associations between stress exposure variables and pCRH levels and rate of rise, adjusting for obstetric/clinical/health related factors. Maternal CTE did not predict pCRH levels at time 1, but positively predicted levels at time 2, and the rate of rise in pCRH across pregnancy. Race did not moderate this association. No additional maternal stress exposures across adulthood or during pregnancy predicted pCRH outcomes. Findings indicate that childhood violence or abuse exposure can become biologically embedded in a manner predicting later prenatal physiology relevant for maternal and offspring health, and that such embedding may be specific to childhood, but not adulthood, stress. Findings also highlight the placental-fetal unit as a mechanistic pathway through which intergenerational transmission of the adverse effects of childhood adversities may occur.},
	language = {en},
	urldate = {2022-04-12},
	journal = {Social Science \& Medicine},
	author = {Steine, Iris M. and LeWinn, Kaja Z. and Lisha, Nadra and Tylavsky, Frances and Smith, Roger and Bowman, Maria and Sathyanarayana, Sheela and Karr, Catherine J. and Smith, Alicia K. and Kobor, Michael and Bush, Nicole R.},
	month = dec,
	year = {2020},
	keywords = {Childhood traumatic events, Maternal psychosocial stress, Placental corticotrophin releasing hormone, Stress during pregnancy},
	pages = {113461},
}

@article{gangestad_function_2012,
	title = {On the function of placental corticotropin-releasing hormone: a role in maternal-fetal conflicts over blood glucose concentrations},
	volume = {87},
	issn = {1469-185X},
	shorttitle = {On the function of placental corticotropin-releasing hormone},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-185X.2012.00226.x},
	doi = {10.1111/j.1469-185X.2012.00226.x},
	abstract = {Throughout the second and third trimesters, the human placenta (and the placenta in other anthropoid primates) produces substantial quantities of corticotropin-releasing hormone (placental CRH), most of which is secreted into the maternal bloodstream. During pregnancy, CRH concentrations rise over 1000-fold. The advantages that led selection to favour placental CRH production and secretion are not yet fully understood. Placental CRH stimulates the production of maternal adrenocorticotropin hormone (ACTH) and cortisol, leading to substantial increases in maternal serum cortisol levels during the third trimester. These effects are puzzling in light of widespread theory that cortisol has harmful effects on the fetus. The maternal hypothalamic-pituitary-adrenal (HPA) axis becomes less sensitive to cortisol during pregnancy, purportedly to protect the fetus from cortisol exposure. Researchers, then, have often looked for beneficial effects of placental CRH that involve receptors outside the HPA system, such as the uterine myometrium (e.g. the placental clock hypothesis). An alternative view is proposed here: the beneficial effect of placental CRH to the fetus lies in the fact that it does stimulate the production of cortisol, which, in turn, leads to greater concentrations of glucose in the maternal bloodstream available for fetal consumption. In this view, maternal HPA insensitivity to placental CRH likely reflects counter-adaptation, as the optimal rate of cortisol production for the fetus exceeds that for the mother. Evidence pertaining to this proposal is reviewed.},
	language = {en},
	number = {4},
	urldate = {2022-04-12},
	journal = {Biological Reviews},
	author = {Gangestad, Steven W. and Caldwell Hooper, Ann E. and Eaton, Melissa A.},
	year = {2012},
	keywords = {corticotropin releasing hormone, cortisol, fetal programming, maternal-fetal conflict, placental hormones},
	pages = {856--873},
}

@misc{us_bureau_of_economic_analysis_real_1947,
	title = {Real {Gross} {Domestic} {Product}},
	shorttitle = {{GDPC1}},
	url = {https://fred.stlouisfed.org/series/GDPC1},
	abstract = {BEA Account Code: A191RX

Real gross domestic product is the inflation adjusted value of the goods and services produced by labor and property located in the United States.For more information see the Guide to the National Income and Product Accounts of the United States (NIPA). For more information, please visit the Bureau of Economic Analysis (http://www.bea.gov/national/pdf/nipaguid.pdf).},
	urldate = {2024-05-20},
	journal = {FRED, Federal Reserve Bank of St. Louis},
	author = {{U.S. Bureau of Economic Analysis}},
	month = jan,
	year = {1947},
	note = {Publisher: FRED, Federal Reserve Bank of St. Louis},
}

@incollection{fritzsche_inductive_2008,
	title = {Inductive {Conformal} {Prediction}: {Theory} and {Application} to {Neural} {Networks}},
	isbn = {978-953-7619-03-9},
	shorttitle = {Inductive {Conformal} {Prediction}},
	url = {http://www.intechopen.com/books/tools_in_artificial_intelligence/inductive_conformal_prediction__theory_and_application_to_neural_networks},
	language = {en},
	urldate = {2024-04-23},
	booktitle = {Tools in {Artificial} {Intelligence}},
	publisher = {InTech},
	author = {Papadopoulos, Harris},
	editor = {Fritzsche, Paula},
	month = aug,
	year = {2008},
	doi = {10.5772/6078},
}


@inproceedings{immer_effective_2023,
	title = {Effective {Bayesian} {Heteroscedastic} {Regression} with {Deep} {Neural} {Networks}},
	abstract = {Flexibly quantifying both irreducible aleatoric and model-dependent epistemic uncertainties plays an important role for complex regression problems. While deep neural networks in principle can provide this flexibility and learn heteroscedastic aleatoric uncertainties through non-linear functions, recent works highlight that maximizing the log likelihood objective parameterized by mean and variance can lead to compromised mean fits since the gradient are scaled by the predictive variance, and propose adjustments in line with this premise. We instead propose to use the natural parametrization of the Gaussian, which has been shown to be more stable for heteroscedastic regression based on non-linear feature maps and Gaussian processes. Further, we emphasize the significance of principled regularization of the network parameters and prediction. We therefore propose an efficient Laplace approximation for heteroscedastic neural networks that allows automatic regularization through empirical Bayes and provides epistemic uncertainties, both of which improve generalization. We showcase on a range of regression problems—including a new heteroscedastic image regression benchmark—that our methods are scalable, improve over previous approaches for heteroscedastic regression, and provide epistemic uncertainty without requiring hyperparameter tuning.},
	language = {en},
	author = {Immer, Alexander and Palumbo, Emanuele and Marx, Alexander and Vogt, Julia E},
	year = {2023},
}

@misc{sluijterman_optimal_2023,
	title = {Optimal {Training} of {Mean} {Variance} {Estimation} {Neural} {Networks}},
	url = {http://arxiv.org/abs/2302.08875},
	abstract = {This paper focusses on the optimal implementation of a Mean Variance Estimation network (MVE network) (Nix and Weigend, 1994). This type of network is often used as a building block for uncertainty estimation methods in a regression setting, for instance Concrete dropout (Gal et al., 2017) and Deep Ensembles (Lakshminarayanan et al., 2017). Specifically, an MVE network assumes that the data is produced from a normal distribution with a mean function and variance function. The MVE network outputs a mean and variance estimate and optimizes the network parameters by minimizing the negative loglikelihood. In our paper, we present two significant insights. Firstly, the convergence difficulties reported in recent work can be relatively easily prevented by following the simple yet often overlooked recommendation from the original authors that a warm-up period should be used. During this period, only the mean is optimized with a fixed variance. We demonstrate the effectiveness of this step through experimentation, highlighting that it should be standard practice. As a sidenote, we examine whether, after the warm-up, it is beneficial to fix the mean while optimizing the variance or to optimize both simultaneously. Here, we do not observe a substantial difference. Secondly, we introduce a novel improvement of the MVE network: separate regularization of the mean and the variance estimate. We demonstrate, both on toy examples and on a number of benchmark UCI regression data sets, that following the original recommendations and the novel separate regularization can lead to significant improvements.},
	urldate = {2024-02-07},
	publisher = {arXiv},
	author = {Sluijterman, Laurens and Cator, Eric and Heskes, Tom},
	month = aug,
	year = {2023},
	note = {arXiv:2302.08875 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{dherin_why_2022,
	title = {Why neural networks ﬁnd simple solutions: the many regularizers of geometric complexity},
	abstract = {In many contexts, simpler models are preferable to more complex models and the control of this model complexity is the goal for many methods in machine learning such as regularization, hyperparameter tuning and architecture design. In deep learning, it has been difﬁcult to understand the underlying mechanisms of complexity control, since many traditional measures are not naturally suitable for deep neural networks. Here we develop the notion of geometric complexity, which is a measure of the variability of the model function, computed using a discrete Dirichlet energy. Using a combination of theoretical arguments and empirical results, we show that many common training heuristics such as parameter norm regularization, spectral norm regularization, ﬂatness regularization, implicit gradient regularization, noise regularization and the choice of parameter initialization all act to control geometric complexity, providing a unifying framework in which to characterize the behavior of deep learning models.},
	language = {en},
	author = {Dherin, Benoit and Munn, Michael and Rosca, Mihaela and Barrett, David G T},
	year = {2022},
}

@article{bishop_training_1995,
	title = {Training with {Noise} is {Equivalent} to {Tikhonov} {Regularization}},
	volume = {7},
	issn = {0899-7667, 1530-888X},
	url = {https://direct.mit.edu/neco/article/7/1/108-116/5828},
	doi = {10.1162/neco.1995.7.1.108},
	abstract = {It is well known that the addition of noise to the input data of a neural network during training can, in some circumstances, lead to signiﬁcant improvements in generalization performance. Previous work has shown that such training with noise is equivalent to a form of regularization in which an extra term is added to the error function. However, the regularization term, which involves second derivatives of the error function, is not bounded below, and so can lead to diﬃculties if used directly in a learning algorithm based on error minimization. In this paper we show that, for the purposes of network training, the regularization term can be reduced to a positive deﬁnite form which involves only ﬁrst derivatives of the network mapping. For a sum-of-squares error function, the regularization term belongs to the class of generalized Tikhonov regularizers. Direct minimization of the regularized error function provides a practical alternative to training with noise.},
	language = {en},
	number = {1},
	urldate = {2023-10-24},
	journal = {Neural Computation},
	author = {Bishop, Chris M.},
	month = jan,
	year = {1995},
	pages = {108--116},
}

@article{hanin_how_nodate,
	title = {How {Data} {Augmentation} affects {Optimization} for {Linear} {Regression}},
	abstract = {Though data augmentation has rapidly emerged as a key tool for optimization in modern machine learning, a clear picture of how augmentation schedules affect optimization and interact with optimization hyperparameters such as learning rate is nascent. In the spirit of classical convex optimization and recent work on implicit bias, the present work analyzes the effect of augmentation on optimization in the simple convex setting of linear regression with MSE loss.},
	language = {en},
	author = {Hanin, Boris and Sun, Yi},
}

@misc{venkatesh_heteroscedastic_2019,
	title = {Heteroscedastic {Calibration} of {Uncertainty} {Estimators} in {Deep} {Learning}},
	url = {http://arxiv.org/abs/1910.14179},
	abstract = {The role of uncertainty quantification (UQ) in deep learning has become crucial with growing use of predictive models in high-risk applications. Though a large class of methods exists for measuring deep uncertainties, in practice, the resulting estimates are found to be poorly calibrated, thus making it challenging to translate them into actionable insights. A common workaround is to utilize a separate recalibration step, which adjusts the estimates to compensate for the miscalibration. Instead, we propose to repurpose the heteroscedastic regression objective as a surrogate for calibration and enable any existing uncertainty estimator to be inherently calibrated. In addition to eliminating the need for recalibration, this also regularizes the training process. Using regression experiments, we demonstrate the effectiveness of the proposed heteroscedastic calibration with two popular uncertainty estimators.},
	urldate = {2023-10-24},
	publisher = {arXiv},
	author = {Venkatesh, Bindya and Thiagarajan, Jayaraman J.},
	month = oct,
	year = {2019},
	note = {arXiv:1910.14179 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{hastie_ridge_2020,
	title = {Ridge {Regularization}: {An} {Essential} {Concept} in {Data} {Science}},
	volume = {62},
	issn = {0040-1706},
	shorttitle = {Ridge {Regularization}},
	url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9410599/},
	doi = {10.1080/00401706.2020.1791959},
	abstract = {Ridge or more formally ℓ2 regularization shows up in many areas of statistics and machine learning. It is one of those essential devices that any good data scientist needs to master for their craft. In this brief ridge fest, I have collected together some of the magic and beauty of ridge that my colleagues and I have encountered over the past 40 years in applied statistics.},
	number = {4},
	urldate = {2023-10-13},
	journal = {Technometrics : a journal of statistics for the physical, chemical, and engineering sciences},
	author = {Hastie, Trevor},
	year = {2020},
	pmid = {36033922},
	pmcid = {PMC9410599},
	pages = {426--433},
}

@misc{sluijterman_optimal_2023-1,
	title = {Optimal {Training} of {Mean} {Variance} {Estimation} {Neural} {Networks}},
	url = {http://arxiv.org/abs/2302.08875},
	abstract = {This paper focusses on the optimal implementation of a Mean Variance Estimation network (MVE network) (Nix and Weigend, 1994). This type of network is often used as a building block for uncertainty estimation methods in a regression setting, for instance Concrete dropout (Gal et al., 2017) and Deep Ensembles (Lakshminarayanan et al., 2017). Specifically, an MVE network assumes that the data is produced from a normal distribution with a mean function and variance function. The MVE network outputs a mean and variance estimate and optimizes the network parameters by minimizing the negative loglikelihood. In our paper, we present two significant insights. Firstly, the convergence difficulties reported in recent work can be relatively easily prevented by following the simple yet often overlooked recommendation from the original authors that a warm-up period should be used. During this period, only the mean is optimized with a fixed variance. We demonstrate the effectiveness of this step through experimentation, highlighting that it should be standard practice. As a sidenote, we examine whether, after the warm-up, it is beneficial to fix the mean while optimizing the variance or to optimize both simultaneously. Here, we do not observe a substantial difference. Secondly, we introduce a novel improvement of the MVE network: separate regularization of the mean and the variance estimate. We demonstrate, both on toy examples and on a number of benchmark UCI regression data sets, that following the original recommendations and the novel separate regularization can lead to significant improvements.},
	urldate = {2023-10-04},
	publisher = {arXiv},
	author = {Sluijterman, Laurens and Cator, Eric and Heskes, Tom},
	month = aug,
	year = {2023},
	note = {arXiv:2302.08875 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}


@misc{noauthor_parallel_nodate,
	title = {parallel - {Jupyter} {Notebook}},
	url = {http://localhost:6006/notebooks/parallel.ipynb},
	urldate = {2023-09-28},
}


@misc{noauthor_uci_nodate,
	title = {{UCI} {Machine} {Learning} {Repository}},
	urldate = {2023-09-27},
}


@misc{yeh_i-cheng_concrete_2007,
	title = {Concrete {Compressive} {Strength}},
	doi = {10.24432/C5PK67},
	publisher = {UCI Machine Learning Repository},
	author = {Yeh, I-Cheng},
	year = {2007},
}


@inproceedings{liu_simple_2020,
	title = {Simple and {Principled} {Uncertainty} {Estimation} with {Deterministic} {Deep} {Learning} via {Distance} {Awareness}},
	language = {en},
	author = {Liu, Jeremiah Zhe and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax-Weiss, Tania and Lakshminarayanan, Balaji},
	year = {2020},
}

@inproceedings{liu_simple_2020-1,
	title = {Simple and {Principled} {Uncertainty} {Estimation} with {Deterministic} {Deep} {Learning} via {Distance} {Awareness}},
	volume = {33},
	url = {https://proceedings.neurips.cc/paper/2020/hash/543e83748234f7cbab21aa0ade66565f-Abstract.html},
	abstract = {Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model.  However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model’s ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer.  On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.},
	urldate = {2023-09-27},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Liu, Jeremiah and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax Weiss, Tania and Lakshminarayanan, Balaji},
	year = {2020},
	pages = {7498--7512},
}


@article{noauthor_variational_2022,
	title = {Variational {Variance}: {Simple} and {Reliable} {Noise} {Variance} {Parameterization}},
	shorttitle = {Variational {Variance}},
	abstract = {Models employing heteroscedastic Gaussian likelihoods parameterized by amortized mean and variance networks are both probabilistically interpretable and highly flexible, but unfortunately can be brittle to optimize. Maximizing log likelihood encourages local Dirac densities for sufficiently flexible mean and variance networks. Data lacking nearby neighbors can provide this flexibility. Gradients near these unbounded optima explode, prohibiting convergence of the mean and thus requiring high noise variance to explain the dependent variable. We propose posterior predictive checks to identify such failures, which we observe can surreptitiously occur alongside high model likelihoods. We find existing approaches that bolster optimization of mean and variance networks to improve likelihoods still exhibit poor predictive mean and variance calibrations. Our notably simpler solution, to treat heteroscedastic variance variationally in an Empirical Bayes regime, regularizes variance away from zero and stabilizes optimization, allowing us to preserve or outperform existing likelihoods while improving predictive mean and variance calibrations and thereby sample quality. We empirically demonstrate these findings on a variety of regression and variational autoencoding tasks.},
	language = {en},
	urldate = {2023-01-30},
	journal = {Transactions on Machine Learning Research},
	month = sep,
	year = {2022},
}

@article{chesher_finite-sample_1991,
	title = {The finite-sample distributions of heteroskedasticity robust {Wald} statistics},
	volume = {47},
	issn = {0304-4076},
	doi = {10.1016/0304-4076(91)90082-O},
	abstract = {The Imhof procedure is used to calculate the exact finite-sample distributions of alternative heteroskedasticity robust Wald-type tests of scalar linear hypothesis in the normal linear model. The tests are distinguished by the variance estimator that they use. These include the White-Eicker and jackknife estimators. The calculations demonstrate the importance of regression design in determining the quality of conventional first-order asymptotic approximations to the null distributions of the tests.},
	language = {en},
	number = {1},
	journal = {Journal of Econometrics},
	author = {Chesher, Andrew and Austin, Gerard},
	month = jan,
	year = {1991},
	pages = {153--173},
}

@misc{liu_simple_2020-2,
	title = {Simple and {Principled} {Uncertainty} {Estimation} with {Deterministic} {Deep} {Learning} via {Distance} {Awareness}},
	abstract = {Bayesian neural networks (BNN) and deep ensembles are principled approaches to estimate the predictive uncertainty of a deep learning model. However their practicality in real-time, industrial-scale applications are limited due to their heavy memory and inference cost. This motivates us to study principled approaches to high-quality uncertainty estimation that require only a single deep neural network (DNN). By formalizing the uncertainty quantification as a minimax learning problem, we first identify input distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data in the input space, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs, by adding a weight normalization step during training and replacing the output layer with a Gaussian process. On a suite of vision and language understanding tasks and on modern architectures (Wide-ResNet and BERT), SNGP is competitive with deep ensembles in prediction, calibration and out-of-domain detection, and outperforms the other single-model approaches.},
	urldate = {2023-02-16},
	publisher = {arXiv},
	author = {Liu, Jeremiah Zhe and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax-Weiss, Tania and Lakshminarayanan, Balaji},
	month = oct,
	year = {2020},
	note = {arXiv:2006.10108 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{doyle_random_2000,
	title = {Random {Walks} and {Electric} {Networks}},
	abstract = {A popular account of the connection between random walks and electric networks.},
	urldate = {2023-02-16},
	publisher = {arXiv},
	author = {Doyle, Peter G. and Snell, J. Laurie},
	month = jan,
	year = {2000},
	note = {arXiv:math/0001057},
	keywords = {Mathematics - Probability},
}


@misc{guo_calibration_2017,
	title = {On {Calibration} of {Modern} {Neural} {Networks}},
	doi = {10.48550/arXiv.1706.04599},
	abstract = {Confidence calibration -- the problem of predicting probability estimates representative of the true correctness likelihood -- is important for classification models in many applications. We discover that modern neural networks, unlike those from a decade ago, are poorly calibrated. Through extensive experiments, we observe that depth, width, weight decay, and Batch Normalization are important factors influencing calibration. We evaluate the performance of various post-processing calibration methods on state-of-the-art architectures with image and document classification datasets. Our analysis and experiments not only offer insights into neural network learning, but also provide a simple and straightforward recipe for practical settings: on most datasets, temperature scaling -- a single-parameter variant of Platt Scaling -- is surprisingly effective at calibrating predictions.},
	urldate = {2023-02-10},
	publisher = {arXiv},
	author = {Guo, Chuan and Pleiss, Geoff and Sun, Yu and Weinberger, Kilian Q.},
	month = aug,
	year = {2017},
	note = {arXiv:1706.04599 [cs]},
	keywords = {Computer Science - Machine Learning},
}

@misc{cao_heteroskedastic_2021,
	title = {Heteroskedastic and {Imbalanced} {Deep} {Learning} with {Adaptive} {Regularization}},
	abstract = {Real-world large-scale datasets are heteroskedastic and imbalanced -- labels have varying levels of uncertainty and label distributions are long-tailed. Heteroskedasticity and imbalance challenge deep learning algorithms due to the difficulty of distinguishing among mislabeled, ambiguous, and rare examples. Addressing heteroskedasticity and imbalance simultaneously is under-explored. We propose a data-dependent regularization technique for heteroskedastic datasets that regularizes different regions of the input space differently. Inspired by the theoretical derivation of the optimal regularization strength in a one-dimensional nonparametric classification setting, our approach adaptively regularizes the data points in higher-uncertainty, lower-density regions more heavily. We test our method on several benchmark tasks, including a real-world heteroskedastic and imbalanced dataset, WebVision. Our experiments corroborate our theory and demonstrate a significant improvement over other methods in noise-robust deep learning.},
	publisher = {arXiv},
	author = {Cao, Kaidi and Chen, Yining and Lu, Junwei and Arechiga, Nikos and Gaidon, Adrien and Ma, Tengyu},
	month = mar,
	year = {2021},
	note = {arXiv:2006.15766 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{pratola_heteroscedastic_2020,
	title = {Heteroscedastic {BART} via {Multiplicative} {Regression} {Trees}},
	volume = {29},
	issn = {1061-8600},
	doi = {10.1080/10618600.2019.1677243},
	abstract = {Bayesian additive regression trees (BART) has become increasingly popular as a flexible and scalable nonparametric regression approach for modern applied statistics problems. For the practitioner dealing with large and complex nonlinear response surfaces, its advantages include a matrix-free formulation and the lack of a requirement to prespecify a confining regression basis. Although flexible in fitting the mean, BART has been limited by its reliance on a constant variance error model. Alleviating this limitation, we propose HBART, a nonparametric heteroscedastic elaboration of BART. In BART, the mean function is modeled with a sum of trees, each of which determines an additive contribution to the mean. In HBART, the variance function is further modeled with a product of trees, each of which determines a multiplicative contribution to the variance. Like the mean model, this flexible, multidimensional variance model is entirely nonparametric with no need for the prespecification of a confining basis. Moreover, with this enhancement, HBART can provide insights into the potential relationships of the predictors with both the mean and the variance. Practical implementations of HBART with revealing new diagnostic plots are demonstrated with simulated and real data on used car prices and song year of release. Supplementary materials for this article are available online.},
	number = {2},
	journal = {Journal of Computational and Graphical Statistics},
	author = {Pratola, M. T. and Chipman, H. A. and George, E. I. and McCulloch, R. E.},
	month = apr,
	year = {2020},
	note = {Publisher: Taylor \& Francis
\_eprint: https://doi.org/10.1080/10618600.2019.1677243},
	keywords = {Applied statistical inference, Big data, Nonparametric regression, Uncertainty quantification},
	pages = {405--417},
}


@misc{stirn_faithful_2022,
	title = {Faithful {Heteroscedastic} {Regression} with {Neural} {Networks}},
	abstract = {Heteroscedastic regression models a Gaussian variable's mean and variance as a function of covariates. Parametric methods that employ neural networks for these parameter maps can capture complex relationships in the data. Yet, optimizing network parameters via log likelihood gradients can yield suboptimal mean and uncalibrated variance estimates. Current solutions side-step this optimization problem with surrogate objectives or Bayesian treatments. Instead, we make two simple modifications to optimization. Notably, their combination produces a heteroscedastic model with mean estimates that are provably as accurate as those from its homoscedastic counterpart (i.e.{\textasciitilde}fitting the mean under squared error loss). For a wide variety of network and task complexities, we find that mean estimates from existing heteroscedastic solutions can be significantly less accurate than those from an equivalently expressive mean-only model. Our approach provably retains the accuracy of an equally flexible mean-only model while also offering best-in-class variance calibration. Lastly, we show how to leverage our method to recover the underlying heteroscedastic noise variance.},
	language = {en},
	urldate = {2023-05-12},
	journal = {arXiv.org},
	author = {Stirn, Andrew and Wessels, Hans-Hermann and Schertzer, Megan and Pereira, Laura and Sanjana, Neville E. and Knowles, David A.},
	month = dec,
	year = {2022},
}

@misc{noauthor_extra_nodate,
	title = {Extra {Materials} {Archive} {Search}},
	urldate = {2023-05-16},
}



@inproceedings{damianou_deep_2013,
	title = {Deep {Gaussian} {Processes}},
	abstract = {In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples.},
	language = {en},
	urldate = {2023-01-30},
	booktitle = {Proceedings of the {Sixteenth} {International} {Conference} on {Artificial} {Intelligence} and {Statistics}},
	publisher = {PMLR},
	author = {Damianou, Andreas and Lawrence, Neil D.},
	month = apr,
	year = {2013},
	note = {ISSN: 1938-7228},
	pages = {207--215},
}

@misc{smith_cyclical_2017,
	title = {Cyclical {Learning} {Rates} for {Training} {Neural} {Networks}},
	abstract = {It is known that the learning rate is the most important hyper-parameter to tune for training deep neural networks. This paper describes a new method for setting the learning rate, named cyclical learning rates, which practically eliminates the need to experimentally find the best values and schedule for the global learning rates. Instead of monotonically decreasing the learning rate, this method lets the learning rate cyclically vary between reasonable boundary values. Training with cyclical learning rates instead of fixed values achieves improved classification accuracy without a need to tune and often in fewer iterations. This paper also describes a simple way to estimate "reasonable bounds" -- linearly increasing the learning rate of the network for a few epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10 and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets, and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These are practical tools for everyone who trains neural networks.},
	urldate = {2023-03-01},
	publisher = {arXiv},
	author = {Smith, Leslie N.},
	month = apr,
	year = {2017},
	note = {arXiv:1506.01186 [cs]},
	keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing},
}

@inproceedings{collier_correlated_2021,
	address = {Nashville, TN, USA},
	title = {Correlated {Input}-{Dependent} {Label} {Noise} in {Large}-{Scale} {Image} {Classification}},
	isbn = {978-1-66544-509-2},
	doi = {10.1109/CVPR46437.2021.00160},
	abstract = {Large scale image classiﬁcation datasets often contain noisy labels. We take a principled probabilistic approach to modelling input-dependent, also known as heteroscedastic, label noise in these datasets. We place a multivariate Normal distributed latent variable on the ﬁnal hidden layer of a neural network classiﬁer. The covariance matrix of this latent variable, models the aleatoric uncertainty due to label noise. We demonstrate that the learned covariance structure captures known sources of label noise between semantically similar and co-occurring classes. Compared to standard neural network training and other baselines, we show significantly improved accuracy on Imagenet ILSVRC 2012 79.3\% (+ 2.6\%), Imagenet-21k 47.0\% (+ 1.1\%) and JFT 64.7\% (+ 1.6\%). We set a new state-of-the-art result on WebVision 1.0 with 76.6\% top-1 accuracy. These datasets range from over 1M to over 300M training examples and from 1k classes to more than 21k classes. Our method is simple to use, and we provide an implementation that is a drop-in replacement for the ﬁnal fully-connected layer in a deep classiﬁer.},
	language = {en},
	urldate = {2023-06-20},
	booktitle = {2021 {IEEE}/{CVF} {Conference} on {Computer} {Vision} and {Pattern} {Recognition} ({CVPR})},
	publisher = {IEEE},
	author = {Collier, Mark and Mustafa, Basil and Kokiopoulou, Efi and Jenatton, Rodolphe and Berent, Jesse},
	month = jun,
	year = {2021},
	pages = {1551--1560},
}

@misc{schaeffer_are_2023,
	title = {Are {Emergent} {Abilities} of {Large} {Language} {Models} a {Mirage}?},
	abstract = {Recent work claims that large language models display emergent abilities, abilities not present in smaller-scale models that are present in larger-scale models. What makes emergent abilities intriguing is two-fold: their sharpness, transitioning seemingly instantaneously from not present to present, and their unpredictability, appearing at seemingly unforeseeable model scales. Here, we present an alternative explanation for emergent abilities: that for a particular task and model family, when analyzing fixed model outputs, emergent abilities appear due to the researcher's choice of metric rather than due to fundamental changes in model behavior with scale. Specifically, nonlinear or discontinuous metrics produce apparent emergent abilities, whereas linear or continuous metrics produce smooth, continuous predictable changes in model performance. We present our alternative explanation in a simple mathematical model, then test it in three complementary ways: we (1) make, test and confirm three predictions on the effect of metric choice using the InstructGPT/GPT-3 family on tasks with claimed emergent abilities; (2) make, test and confirm two predictions about metric choices in a meta-analysis of emergent abilities on BIG-Bench; and (3) show to choose metrics to produce never-before-seen seemingly emergent abilities in multiple vision tasks across diverse deep networks. Via all three analyses, we provide evidence that alleged emergent abilities evaporate with different metrics or with better statistics, and may not be a fundamental property of scaling AI models.},
	urldate = {2023-05-25},
	publisher = {arXiv},
	author = {Schaeffer, Rylan and Miranda, Brando and Koyejo, Sanmi},
	month = may,
	year = {2023},
	note = {arXiv:2304.15004 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning},
}


@article{schulz_tutorial_2018,
	title = {A tutorial on {Gaussian} process regression: {Modelling}, exploring, and exploiting functions},
	volume = {85},
	issn = {0022-2496},
	shorttitle = {A tutorial on {Gaussian} process regression},
	doi = {10.1016/j.jmp.2018.03.001},
	abstract = {This tutorial introduces the reader to Gaussian process regression as an expressive tool to model, actively explore and exploit unknown functions. Gaussian process regression is a powerful, non-parametric Bayesian approach towards regression problems that can be utilized in exploration and exploitation scenarios. This tutorial aims to provide an accessible introduction to these techniques. We will introduce Gaussian processes which generate distributions over functions used for Bayesian non-parametric regression, and demonstrate their use in applications and didactic examples including simple regression problems, a demonstration of kernel-encoded prior assumptions and compositions, a pure exploration scenario within an optimal design framework, and a bandit-like exploration–exploitation scenario where the goal is to recommend movies. Beyond that, we describe a situation modelling risk-averse exploration in which an additional constraint (not to sample below a certain threshold) needs to be accounted for. Lastly, we summarize recent psychological experiments utilizing Gaussian processes. Software and literature pointers are also provided.},
	language = {en},
	urldate = {2023-01-30},
	journal = {Journal of Mathematical Psychology},
	author = {Schulz, Eric and Speekenbrink, Maarten and Krause, Andreas},
	month = aug,
	year = {2018},
	keywords = {Active learning, Bandit problems, Exploration–exploitation, Gaussian process regression},
	pages = {1--16},
}

@misc{collier_simple_2020,
	title = {A {Simple} {Probabilistic} {Method} for {Deep} {Classification} under {Input}-{Dependent} {Label} {Noise}},
	abstract = {Datasets with noisy labels are a common occurrence in practical applications of classification methods. We propose a simple probabilistic method for training deep classifiers under input-dependent (heteroscedastic) label noise. We assume an underlying heteroscedastic generative process for noisy labels. To make gradient based training feasible we use a temperature parameterized softmax as a smooth approximation to the assumed generative process. We illustrate that the softmax temperature controls a bias-variance trade-off for the approximation. By tuning the softmax temperature, we improve accuracy, log-likelihood and calibration on both image classification benchmarks with controlled label noise as well as Imagenet-21k which has naturally occurring label noise. For image segmentation, our method increases the mean IoU on the PASCAL VOC and Cityscapes datasets by more than 1\% over the state-of-the-art model.},
	urldate = {2023-01-30},
	publisher = {arXiv},
	author = {Collier, Mark and Mustafa, Basil and Kokiopoulou, Efi and Jenatton, Rodolphe and Berent, Jesse},
	month = nov,
	year = {2020},
	note = {arXiv:2003.06778 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{collier_simple_2020-1,
	title = {A {Simple} {Probabilistic} {Method} for {Deep} {Classification} under {Input}-{Dependent} {Label} {Noise}},
	abstract = {Datasets with noisy labels are a common occurrence in practical applications of classification methods. We propose a simple probabilistic method for training deep classifiers under input-dependent (heteroscedastic) label noise. We assume an underlying heteroscedastic generative process for noisy labels. To make gradient based training feasible we use a temperature parameterized softmax as a smooth approximation to the assumed generative process. We illustrate that the softmax temperature controls a bias-variance trade-off for the approximation. By tuning the softmax temperature, we improve accuracy, log-likelihood and calibration on both image classification benchmarks with controlled label noise as well as Imagenet-21k which has naturally occurring label noise. For image segmentation, our method increases the mean IoU on the PASCAL VOC and Cityscapes datasets by more than 1\% over the state-of-the-art model.},
	urldate = {2023-01-30},
	publisher = {arXiv},
	author = {Collier, Mark and Mustafa, Basil and Kokiopoulou, Efi and Jenatton, Rodolphe and Berent, Jesse},
	month = nov,
	year = {2020},
	note = {arXiv:2003.06778 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{parker_general_2021,
	title = {A general {Bayesian} model for heteroskedastic data with fully conjugate full-conditional distributions},
	volume = {91},
	issn = {0094-9655},
	doi = {10.1080/00949655.2021.1925279},
	abstract = {Models for heteroskedastic data are relevant in a variety of applications ranging from financial time series to environmental statistics. However, the topic of modelling the variance function conditionally has not seen as much attention as modelling the mean. Volatility models have been used in specific applications, but these models can be difficult to fit in a Bayesian setting due to posterior distributions that are challenging to sample efficiently. In this work, we introduce a general model for heteroskedastic data. This approach models the conditional variance as a function of any desired covariates or random effects. We rely on multivariate log-Gamma distribution theory to construct priors that yield fully conjugate full-conditional distributions for Gibbs sampling. Furthermore, we extend the model to a deep learning approach that can provide highly accurate estimates for time dependent data. We also provide an extension for heavy-tailed data. We illustrate our methodology via three applications.},
	number = {15},
	journal = {Journal of Statistical Computation and Simulation},
	author = {Parker, Paul A. and Holan, Scott H. and Wills, Skye A.},
	month = oct,
	year = {2021},
	note = {Publisher: Taylor \& Francis
\_eprint: https://doi.org/10.1080/00949655.2021.1925279},
	keywords = {Deep learning, Gibbs sampling, echo state network, mixed models, multivariate log-Gamma, spatial, volatility},
	pages = {3207--3227},
}

@article{hinne_conceptual_2020,
	title = {A {Conceptual} {Introduction} to {Bayesian} {Model} {Averaging}},
	volume = {3},
	issn = {2515-2459},
	doi = {10.1177/2515245919898657},
	abstract = {Many statistical scenarios initially involve several candidate models that describe the data-generating process. Analysis often proceeds by first selecting the best model according to some criterion and then learning about the parameters of this selected model. Crucially, however, in this approach the parameter estimates are conditioned on the selected model, and any uncertainty about the model-selection process is ignored. An alternative is to learn the parameters for all candidate models and then combine the estimates according to the posterior probabilities of the associated models. This approach is known as Bayesian model averaging (BMA). BMA has several important advantages over all-or-none selection methods, but has been used only sparingly in the social sciences. In this conceptual introduction, we explain the principles of BMA, describe its advantages over all-or-none model selection, and showcase its utility in three examples: analysis of covariance, meta-analysis, and network analysis.},
	number = {2},
	journal = {Advances in Methods and Practices in Psychological Science},
	author = {Hinne, Max and Gronau, Quentin F. and van den Bergh, Don and Wagenmakers, Eric-Jan},
	month = jun,
	year = {2020},
	note = {Publisher: SAGE Publications Inc},
	pages = {200--215},
}



@misc{noauthor_uci_nodate,
	title = {{UCI} {Machine} {Learning} {Repository}: {Yacht} {Hydrodynamics} {Data} {Set}},
	url = {https://archive.ics.uci.edu/ml/datasets/yacht+hydrodynamics},
	urldate = {2023-05-17},
}

@misc{noauthor_uci_nodate-1,
	title = {{UCI} {Machine} {Learning} {Repository}},
	url = {https://archive.ics.uci.edu/ml/index.php},
	urldate = {2023-05-07},
}

@inproceedings{krizhevsky_imagenet_2012,
	title = {{ImageNet} {Classification} with {Deep} {Convolutional} {Neural} {Networks}},
	volume = {25},
	url = {https://papers.nips.cc/paper/2012/hash/c399862d3b9d6b76c8436e924a68c45b-Abstract.html},
	abstract = {We trained a large, deep convolutional neural network to classify the 1.3 million high-resolution images in the LSVRC-2010 ImageNet training set into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 39.7{\textbackslash}\% and 18.9{\textbackslash}\% which is considerably better than the previous state-of-the-art results. The neural network, which has 60 million parameters and 500,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and two globally connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of convolutional nets. To reduce overfitting in the globally connected layers we employed a new regularization method that proved to be very effective.},
	urldate = {2023-02-17},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Krizhevsky, Alex and Sutskever, Ilya and Hinton, Geoffrey E},
	year = {2012},
}

@article{olvera_astivia_heteroskedasticity_nodate,
	title = {Heteroskedasticity in {Multiple} {Regression} {Analysis}: {What} it is, {How} to {Detect} it and {How} to {Solve} it with {Applications} in {R} and {SPSS}},
	shorttitle = {Heteroskedasticity in {Multiple} {Regression} {Analysis}},
	url = {https://scholarworks.umass.edu/pare/vol24/iss1/1/},
	doi = {10.7275/Q5XR-FR95},
	language = {en},
	urldate = {2023-02-17},
	author = {Olvera Astivia, Oscar L and Zumbo, Bruno D},
	note = {Publisher: University of Massachusetts Amherst},
}

@incollection{ahmad_deep_2019,
	address = {Singapore},
	series = {{SpringerBriefs} in {Computer} {Science}},
	title = {Deep {Learning} {Methods} and {Applications}},
	isbn = {9789811334597},
	url = {https://doi.org/10.1007/978-981-13-3459-7_3},
	abstract = {This chapter introduces the various methods existing beneath the umbrella of deep learning paradigm, their intricate details, and their applications in various fields. Deep learning has substantially improved the predictive capacity of computing devices, due to the availability of big data, with the help of superior learning algorithms. It has made it possible as well as practical to integrate machine learning with sophisticated applications including image recognition, object detection, self-driving cars, drug discovery, and disease detection. The superior and reliable performance of deep learning methods has attracted the attention of researchers working in every field of science to utilize their strengths in order to solve problems. In addition to that, the knowledge reuse in deep learning is an interesting aspect of this technology which will also be discussed.},
	language = {en},
	urldate = {2023-02-17},
	booktitle = {Deep {Learning}: {Convergence} to {Big} {Data} {Analytics}},
	publisher = {Springer},
	author = {Ahmad, Jamil and Farman, Haleem and Jan, Zahoor},
	editor = {Khan, Murad and Jan, Bilal and Farman, Haleem},
	year = {2019},
	doi = {10.1007/978-981-13-3459-7_3},
	pages = {31--42},
}

@inproceedings{mathew_deep_2021,
	address = {Singapore},
	series = {Advances in {Intelligent} {Systems} and {Computing}},
	title = {Deep {Learning} {Techniques}: {An} {Overview}},
	isbn = {9789811533839},
	shorttitle = {Deep {Learning} {Techniques}},
	doi = {10.1007/978-981-15-3383-9_54},
	abstract = {Deep learning is a class of machine learning whichMathew, Amitha performsAmudha, P. muchSivakumari, S. better on unstructured data. Deep learning techniques are outperforming current machine learning techniques. It enables computational models to learn features progressively from data at multiple levels. The popularity of deep learning amplified as the amount of data available increased as well as the advancement of hardware that provides powerful computers. This article comprises the evolution of deep learning, various approaches to deep learning, architectures of deep learning, methods, and applications.},
	language = {en},
	booktitle = {Advanced {Machine} {Learning} {Technologies} and {Applications}},
	publisher = {Springer},
	author = {Mathew, Amitha and Amudha, P. and Sivakumari, S.},
	editor = {Hassanien, Aboul Ella and Bhatnagar, Roheet and Darwish, Ashraf},
	year = {2021},
	keywords = {Convolutional neural network (CNN), Deep belief network (DBN), Deep learning (DL), Generative adversarial network (GAN), Recurrent neural network (RNN)},
	pages = {599--608},
}

@article{altl_condensed_nodate,
	title = {Condensed {Matter} {Field} {Theory}, {Second} {Edition}},
	language = {en},
	journal = {Condensed Matter Field Theory},
	author = {Altl, Alex and er, Simons Ben D},
}

@article{altl_condensed_nodate-1,
	title = {Condensed {Matter} {Field} {Theory}, {Second} {Edition}},
	language = {en},
	journal = {Condensed Matter Field Theory},
	author = {Altl, Alex and er, Simons Ben D},
}

@inproceedings{goldberg_regression_1997,
	title = {Regression with {Input}-dependent {Noise}: {A} {Gaussian} {Process} {Treatment}},
	volume = {10},
	shorttitle = {Regression with {Input}-dependent {Noise}},
	url = {https://proceedings.neurips.cc/paper/1997/hash/afe434653a898da20044041262b3ac74-Abstract.html},
	abstract = {Gaussian processes  provide natural non-parametric prior distribu(cid:173) tions over regression functions.  In this paper we consider regression  problems where there is  noise  on  the output,  and  the variance  of  the  noise  depends  on  the  inputs.  If we  assume  that  the  noise  is  a  smooth  function  of the  inputs,  then  it  is  natural  to  model  the  noise variance using a second Gaussian process,  in addition to the  Gaussian process governing the noise-free output  value.  We  show  that prior uncertainty about the parameters controlling both pro(cid:173) cesses  can  be  handled  and  that  the  posterior  distribution  of  the  noise  rate can  be  sampled from  using  Markov  chain Monte  Carlo  methods.  Our results on a synthetic data set give a posterior noise  variance that well-approximates the true variance.},
	urldate = {2023-02-11},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {MIT Press},
	author = {Goldberg, Paul and Williams, Christopher and Bishop, Christopher},
	year = {1997},
}

@book{rasmussen_gaussian_2006,
	address = {Cambridge, Mass},
	series = {Adaptive computation and machine learning},
	title = {Gaussian processes for machine learning},
	isbn = {978-0-262-18253-9},
	language = {en},
	publisher = {MIT Press},
	author = {Rasmussen, Carl Edward and Williams, Christopher K. I.},
	year = {2006},
	note = {OCLC: ocm61285753},
	keywords = {Data processing, Gaussian processes, Machine learning, Mathematical models},
}


@misc{lakshminarayanan_simple_2017-1,
	title = {Simple and {Scalable} {Predictive} {Uncertainty} {Estimation} using {Deep} {Ensembles}},
	url = {http://arxiv.org/abs/1612.01474},
	abstract = {Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs, which learn a distribution over weights, are currently the state-of-the-art for estimating predictive uncertainty; however these require significant modifications to the training procedure and are computationally expensive compared to standard (non-Bayesian) NNs. We propose an alternative to Bayesian NNs that is simple to implement, readily parallelizable, requires very little hyperparameter tuning, and yields high quality predictive uncertainty estimates. Through a series of experiments on classification and regression benchmarks, we demonstrate that our method produces well-calibrated uncertainty estimates which are as good or better than approximate Bayesian NNs. To assess robustness to dataset shift, we evaluate the predictive uncertainty on test examples from known and unknown distributions, and show that our method is able to express higher uncertainty on out-of-distribution examples. We demonstrate the scalability of our method by evaluating predictive uncertainty estimates on ImageNet.},
	publisher = {arXiv},
	author = {Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles},
	month = nov,
	year = {2017},
	note = {arXiv:1612.01474 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{wang_robust_2018,
	title = {Robust {Probabilistic} {Modeling} with {Bayesian} {Data} {Reweighting}},
	url = {http://arxiv.org/abs/1606.03860},
	abstract = {Probabilistic models analyze data by relying on a set of assumptions. Data that exhibit deviations from these assumptions can undermine inference and prediction quality. Robust models offer protection against mismatch between a model's assumptions and reality. We propose a way to systematically detect and mitigate mismatch of a large class of probabilistic models. The idea is to raise the likelihood of each observation to a weight and then to infer both the latent variables and the weights from data. Inferring the weights allows a model to identify observations that match its assumptions and down-weight others. This enables robust inference and improves predictive accuracy. We study four different forms of mismatch with reality, ranging from missing latent groups to structure misspecification. A Poisson factorization analysis of the Movielens 1M dataset shows the benefits of this approach in a practical scenario.},
	publisher = {arXiv},
	author = {Wang, Yixin and Kucukelbir, Alp and Blei, David M.},
	month = jun,
	year = {2018},
	note = {arXiv:1606.03860 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{seitzer_pitfalls_2022-1,
	title = {On the {Pitfalls} of {Heteroscedastic} {Uncertainty} {Estimation} with {Probabilistic} {Neural} {Networks}},
	url = {http://arxiv.org/abs/2203.09168},
	abstract = {Capturing aleatoric uncertainty is a critical part of many machine learning systems. In deep learning, a common approach to this end is to train a neural network to estimate the parameters of a heteroscedastic Gaussian distribution by maximizing the logarithm of the likelihood function under the observed data. In this work, we examine this approach and identify potential hazards associated with the use of log-likelihood in conjunction with gradient-based optimizers. First, we present a synthetic example illustrating how this approach can lead to very poor but stable parameter estimates. Second, we identify the culprit to be the log-likelihood loss, along with certain conditions that exacerbate the issue. Third, we present an alternative formulation, termed \${\textbackslash}beta\$-NLL, in which each data point's contribution to the loss is weighted by the \${\textbackslash}beta\$-exponentiated variance estimate. We show that using an appropriate \${\textbackslash}beta\$ largely mitigates the issue in our illustrative example. Fourth, we evaluate this approach on a range of domains and tasks and show that it achieves considerable improvements and performs more robustly concerning hyperparameters, both in predictive RMSE and log-likelihood criteria.},
	publisher = {arXiv},
	author = {Seitzer, Maximilian and Tavakoli, Arash and Antic, Dimitrije and Martius, Georg},
	month = apr,
	year = {2022},
	note = {arXiv:2203.09168 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}
