
@inproceedings{sun2018functional,
  title     = {{FUNCTIONAL} {VARIATIONAL} {BAYESIAN} {NEURAL} {NETWORKS}},
  author    = {Shengyang Sun and Guodong Zhang and Jiaxin Shi and Roger Grosse},
  booktitle = {International Conference on Learning Representations},
  year      = {2019},
  url       = {https://openreview.net/forum?id=rkxacs0qY7}
}


@inproceedings{rudner2022fsvi,
  title   = {{T}ractabe {F}unction-{S}pace {V}ariational {I}nference in {B}ayesian {N}eural {N}etworks},
  author  = {Tim G. J. Rudner and Zonghao Chen and Yee Whye Teh and Yarin Gal},
  booktitle = {Advances in Neural Information Processing Systems},
  year    = {2022}
}


@misc{quang2022gpkl,
  title         = {Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings},
  author        = {Minh Ha Quang},
  year          = {2022},
  eprint        = {2207.08406},
  archiveprefix = {arXiv},
  primaryclass  = {stat.ML}
}

@article{quang2019regularizedKL,
  title={Regularized Divergences Between Covariance Operators and Gaussian Measures on Hilbert Spaces},
  author={Minh Ha Quang},
  journal={Journal of Theoretical Probability},
  year={2019},
  url={https://api.semanticscholar.org/CorpusID:118638845}
}

@inproceedings{burt2020understanding,
author = {Wild, Veit D. and Hu, Robert and Sejdinovic, Dino},
title = {Generalized variational inference in function spaces: Gaussian measures meet bayesian deep learning},
year = {2022},
isbn = {9781713871088},
publisher = {Curran Associates Inc.},
address = {Red Hook, NY, USA},
abstract = {We develop a framework for generalized variational inference in infinite- dimensional function spaces and use it to construct a method termed Gaussian Wasserstein inference (GWI). GWI leverages the Wasserstein distance between Gaussian measures on the Hilbert space of square-integrable functions in order to determine a variational posterior using a tractable optimization criterion. It avoids pathologies arising in standard variational function space inference. An exciting application of GWI is the ability to use deep neural networks in the variational parametrization of GWI, combining their superior predictive performance with the principled uncertainty quantification analogous to that of Gaussian processes. The proposed method obtains state-of-the-art performance on several benchmark datasets.},
booktitle = {Proceedings of the 36th International Conference on Neural Information Processing Systems},
articleno = {269},
numpages = {15},
location = {New Orleans, LA, USA},
series = {NIPS '22}
}

@InProceedings{ma2019variational,
  title = 	 {Variational Implicit Processes},
  author =       {Ma, Chao and Li, Yingzhen and Hernandez-Lobato, Jose Miguel},
  booktitle = 	 {Proceedings of the 36th International Conference on Machine Learning},
  pages = 	 {4222--4233},
  year = 	 {2019},
  editor = 	 {Chaudhuri, Kamalika and Salakhutdinov, Ruslan},
  volume = 	 {97},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {09--15 Jun},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v97/ma19b/ma19b.pdf},
  url = 	 {https://proceedings.mlr.press/v97/ma19b.html},
  abstract = 	 {We introduce the implicit processes (IPs), a stochastic process that places implicitly defined multivariate distributions over any finite collections of random variables. IPs are therefore highly flexible implicit priors over <em>functions</em>, with examples including data simulators, Bayesian neural networks and non-linear transformations of stochastic processes. A novel and efficient approximate inference algorithm for IPs, namely the variational implicit processes (VIPs), is derived using generalised wake-sleep updates. This method returns simple update equations and allows scalable hyper-parameter learning with stochastic optimization. Experiments show that VIPs return better uncertainty estimates and lower errors over existing inference methods for challenging models such as Bayesian neural networks, and Gaussian processes.}
}


@inproceedings{ma2021funcVIspg,
 author = {Ma, Chao and Hern\'{a}ndez-Lobato, Jos\'{e} Miguel},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {M. Ranzato and A. Beygelzimer and Y. Dauphin and P.S. Liang and J. Wortman Vaughan},
 pages = {21795--21807},
 publisher = {Curran Associates, Inc.},
 title = {Functional Variational Inference based on Stochastic Process Generators},
 url = {https://proceedings.neurips.cc/paper_files/paper/2021/file/b613e70fd9f59310cf0a8d33de3f2800-Paper.pdf},
 volume = {34},
 year = {2021}
}

@inproceedings{wild2022gvi,
title={Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learning},
author={Veit David Wild and Robert Hu and Dino Sejdinovic},
booktitle={Advances in Neural Information Processing Systems},
editor={Alice H. Oh and Alekh Agarwal and Danielle Belgrave and Kyunghyun Cho},
year={2022},
url={https://openreview.net/forum?id=mMT8bhVBoUa}
}

@article{tran2022NeedGoodfuncpriorforBNNs,
  author  = {Ba-Hien Tran and Simone Rossi and Dimitrios Milios and Maurizio Filippone},
  title   = {All You Need is a Good Functional Prior for Bayesian Deep Learning},
  journal = {Journal of Machine Learning Research},
  year    = {2022},
  volume  = {23},
  number  = {74},
  pages   = {1--56},
  url     = {http://jmlr.org/papers/v23/20-1340.html}
}

@book{williams2006gaussian,
  title     = {Gaussian processes for machine learning},
  author    = {Williams, Christopher KI and Rasmussen, Carl Edward},
  volume    = {2},
  year      = {2006},
  publisher = {MIT press Cambridge, MA}
}

@article{knoblauch2019generalized,
  author  = {Jeremias Knoblauch and Jack Jewson and Theodoros Damoulas},
  title   = {An Optimization-centric View on Bayes' Rule: Reviewing and Generalizing Variational Inference},
  journal = {Journal of Machine Learning Research},
  year    = {2022},
  volume  = {23},
  number  = {132},
  pages   = {1--109},
  url     = {http://jmlr.org/papers/v23/19-1047.html}
}

@inproceedings{rudner2022ContinualLearningFSVI,
  title     = {Continual Learning via Sequential Function-Space Variational Inference},
  author    = {Rudner, Tim G. J. and Bickford Smith, Freddie and Feng, Qixuan and Teh, Yee Whye and Gal, Yarin},
  booktitle = {Proceedings of the 39th International Conference on Machine Learning},
  pages     = {18871--18887},
  year      = {2022},
  editor    = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
  volume    = {162},
  series    = {Proceedings of Machine Learning Research},
  month     = {17--23 Jul},
  publisher = {PMLR},
  pdf       = {https://proceedings.mlr.press/v162/rudner22a/rudner22a.pdf},
  url       = {https://proceedings.mlr.press/v162/rudner22a.html},
  abstract  = {Sequential Bayesian inference over predictive functions is a natural framework for continual learning from streams of data. However, applying it to neural networks has proved challenging in practice. Addressing the drawbacks of existing techniques, we propose an optimization objective derived by formulating continual learning as sequential function-space variational inference. In contrast to existing methods that regularize neural network parameters directly, this objective allows parameters to vary widely during training, enabling better adaptation to new tasks. Compared to objectives that directly regularize neural network predictions, the proposed objective allows for more flexible variational distributions and more effective regularization. We demonstrate that, across a range of task sequences, neural networks trained via sequential function-space variational inference achieve better predictive accuracy than networks trained with related methods while depending less on maintaining a set of representative points from previous tasks.}
}

@InProceedings{immer2021linlaplace,
  title = 	 { Improving predictions of Bayesian neural nets via local linearization },
  author =       {Immer, Alexander and Korzepa, Maciej and Bauer, Matthias},
  booktitle = 	 {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {703--711},
  year = 	 {2021},
  editor = 	 {Banerjee, Arindam and Fukumizu, Kenji},
  volume = 	 {130},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {13--15 Apr},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v130/immer21a/immer21a.pdf},
  url = 	 {https://proceedings.mlr.press/v130/immer21a.html},
  abstract = 	 { The generalized Gauss-Newton (GGN) approximation is often used to make practical Bayesian deep learning approaches scalable by replacing a second order derivative with a product of first order derivatives. In this paper we argue that the GGN approximation should be understood as a local linearization of the underlying Bayesian neural network (BNN), which turns the BNN into a generalized linear model (GLM). Because we use this linearized model for posterior inference, we should also predict using this modified model instead of the original one. We refer to this modified predictive as "GLM predictive" and show that it effectively resolves common underfitting problems of the Laplace approximation. It extends previous results in this vein to general likelihoods and has an equivalent Gaussian process formulation, which enables alternative inference schemes for BNNs in function space. We demonstrate the effectiveness of our approach on several standard classification datasets as well as on out-of-distribution detection. We provide an implementation at https://github.com/AlexImmer/BNN-predictions. }
}


@misc{FlamShepherd2017MappingGP,
  title  = {Mapping Gaussian Process Priors to Bayesian Neural Networks},
  author = {Daniel Flam-Shepherd and James Requeima and David Duvenaud},
  year   = {2017}
}

@inproceedings{pearce2020GPpriorsBNN,
  title     = {Expressive Priors in Bayesian Neural Networks: Kernel Combinations and Periodic Functions},
  author    = {Pearce, Tim and Tsuchida, Russell and Zaki, Mohamed and Brintrup, Alexandra and Neely, Andy},
  booktitle = {Proceedings of The 35th Uncertainty in Artificial Intelligence Conference},
  pages     = {134--144},
  year      = {2020},
  editor    = {Adams, Ryan P. and Gogate, Vibhav},
  volume    = {115},
  series    = {Proceedings of Machine Learning Research},
  month     = {22--25 Jul},
  publisher = {PMLR},
  pdf       = {http://proceedings.mlr.press/v115/pearce20a/pearce20a.pdf},
  url       = {https://proceedings.mlr.press/v115/pearce20a.html},
  abstract  = {A simple, flexible approach to creating expressive priors in Gaussian process (GP) models makes new kernels from a combination of basic kernels, e.g. summing a periodic and linear kernel can capture seasonal variation with a long term trend. Despite a well-studied link between GPs and Bayesian neural networks (BNNs), the BNN analogue of this has not yet been explored. This paper derives BNN architectures mirroring such kernel combinations. Furthermore, it shows how BNNs can produce periodic kernels, which are often useful in this context. These ideas provide a principled approach to designing BNNs that incorporate prior knowledge about a function. We showcase the practical value of these ideas with illustrative experiments in supervised and reinforcement learning settings.}
}

@article{matsubara2022ridgelet,
  author  = {Takuo Matsubara and Chris J. Oates and FranÃ§ois-Xavier Briol},
  title   = {The Ridgelet Prior: A Covariance Function Approach to Prior Specification for Bayesian Neural Networks},
  journal = {Journal of Machine Learning Research},
  year    = {2021},
  volume  = {22},
  number  = {157},
  pages   = {1--57},
  url     = {http://jmlr.org/papers/v22/20-1300.html}
}

@inproceedings{maddox2021FastAdapt,
  title     = { Fast Adaptation with Linearized Neural Networks },
  author    = {Maddox, Wesley and Tang, Shuai and Moreno, Pablo and Gordon Wilson, Andrew and Damianou, Andreas},
  booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics},
  pages     = {2737--2745},
  year      = {2021},
  editor    = {Banerjee, Arindam and Fukumizu, Kenji},
  volume    = {130},
  series    = {Proceedings of Machine Learning Research},
  month     = {13--15 Apr},
  publisher = {PMLR},
  pdf       = {http://proceedings.mlr.press/v130/maddox21a/maddox21a.pdf},
  url       = {https://proceedings.mlr.press/v130/maddox21a.html},
  abstract  = { The inductive biases of trained neural networks are difficult to understand and, consequently, to adapt to new settings. We study the inductive biases of linearizations of neural networks, which we show to be surprisingly good summaries of the full network functions. Inspired by this finding, we propose a technique for embedding these inductive biases into Gaussian processes through a kernel designed from the Jacobian of the network. In this setting, domain adaptation takes the form of interpretable posterior inference, with accompanying uncertainty estimation. This inference is analytic and free of local optima issues found in standard techniques such as fine-tuning neural network weights to a new task. We develop significant computational speed-ups based on matrix multiplies, including a novel implementation for scalable Fisher vector products. Our experiments on both image classification and regression demonstrate the promise and convenience of this framework for transfer learning, compared to neural network fine-tuning. }
}

@article{chen2021gaussian,
author = {Chen, Hao and Zheng, Lili and Al Kontar, Raed and Raskutti, Garvesh},
title = {Gaussian process parameter estimation using mini-batch stochastic gradient descent: convergence guarantees and empirical benefits},
year = {2022},
issue_date = {January 2022},
publisher = {JMLR.org},
volume = {23},
number = {1},
issn = {1532-4435},
abstract = {Stochastic gradient descent (SGD) and its variants have established themselves as the go-to algorithms for large-scale machine learning problems with independent samples due to their generalization performance and intrinsic computational advantage. However, the fact that the stochastic gradient is a biased estimator of the full gradient with correlated samples has led to the lack of theoretical understanding of how SGD behaves under correlated settings and hindered its use in such cases. In this paper, we focus on hyperparmeter estimation for the Gaussian process (GP) and take a step forward towards breaking the barrier by proving minibatch SGD converges to a critical point of the full log-likelihood loss function, and recovers model hyperparameters with rate O(1/K) for K iterations, up to a statistical error term depending on the minibatch size. Our theoretical guarantees hold provided that the kernel functions exhibit exponential or polynomial eigendecay which is satisfied by a wide range of kernels commonly used in GPs. Numerical studies on both simulated and real datasets demonstrate that minibatch SGD has better generalization over state-of-the-art GP methods while reducing the computational burden and opening a new, previously unexplored, data size regime for GPs.},
journal = {J. Mach. Learn. Res.},
month = jan,
articleno = {227},
numpages = {59},
keywords = {computational speed-up, convergence rate, Gaussian processes, stochastic optimization}
}

@InProceedings{nalisnick2020predictive,
  title = 	 { Predictive Complexity Priors },
  author =       {Nalisnick, Eric and Gordon, Jonathan and Miguel Hernandez-Lobato, Jose},
  booktitle = 	 {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {694--702},
  year = 	 {2021},
  editor = 	 {Banerjee, Arindam and Fukumizu, Kenji},
  volume = 	 {130},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {13--15 Apr},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v130/nalisnick21a/nalisnick21a.pdf},
  url = 	 {https://proceedings.mlr.press/v130/nalisnick21a.html},
  abstract = 	 { Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and uninformative can have unintuitive and detrimental effects on a model’s predictions. For this reason, we propose predictive complexity priors: a functional prior that is defined by comparing the model’s predictions to those of a reference model. Although originally defined on the model outputs, we transfer the prior to the model parameters via a change of variables. The traditional Bayesian workflow can then proceed as usual. We apply our predictive complexity prior to high-dimensional regression, reasoning over neural network depth, and sharing of statistical strength for few-shot learning. }
}

@misc{simpson2022,
    author = {Dan Simpson},
    editor = {},
    title  = {Priors for the Parameters in a {Gaussian} Process},
    date   = {2022-09-27},
    year  = {2022},
    url    = {https://dansblog.netlify.app/posts/2022-09-07-priors5/priors5.html},
    langid = {en}
}

@InProceedings{kerrigan2023diffusion,
  title = 	 {Diffusion Generative Models in Infinite Dimensions},
  author =       {Kerrigan, Gavin and Ley, Justin and Smyth, Padhraic},
  booktitle = 	 {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {9538--9563},
  year = 	 {2023},
  editor = 	 {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
  volume = 	 {206},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {25--27 Apr},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v206/kerrigan23a/kerrigan23a.pdf},
  url = 	 {https://proceedings.mlr.press/v206/kerrigan23a.html},
  abstract = 	 {Diffusion generative models have recently been applied to domains where the available data can be seen as a discretization of an underlying function, such as audio signals or time series. However, these models operate directly on the discretized data, and there are no semantics in the modeling process that relate the observed data to the underlying functional forms. We generalize diffusion models to operate directly in function space by developing the foundational theory for such models in terms of Gaussian measures on Hilbert spaces. A significant benefit of our function space point of view is that it allows us to explicitly specify the space of functions we are working in, leading us to develop methods for diffusion generative modeling in Sobolev spaces. Our approach allows us to perform both unconditional and conditional generation of function-valued data. We demonstrate our methods on several synthetic and real-world benchmarks.}
}


@article{Santin2016approxEigenfunc,
  doi       = {10.1007/s10444-015-9449-5},
  url       = {https://doi.org/10.1007%2Fs10444-015-9449-5},
  year      = 2016,
  month     = {jan},
  publisher = {Springer Science and Business Media {LLC}},
  volume    = {42},
  number    = {4},
  pages     = {973--993},
  author    = {G. Santin and R. Schaback},
  title     = {Approximation of eigenfunctions in kernel-based spaces},
  journal   = {Advances in Computational Mathematics}
}

@InProceedings{belkin2018approximation,
  title = 	 {Approximation beats concentration? An approximation view  on inference with smooth radial kernels},
  author =       {Belkin, Mikhail},
  booktitle = 	 {Proceedings of the 31st  Conference On Learning Theory},
  pages = 	 {1348--1361},
  year = 	 {2018},
  editor = 	 {Bubeck, Sébastien and Perchet, Vianney and Rigollet, Philippe},
  volume = 	 {75},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {06--09 Jul},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v75/belkin18a/belkin18a.pdf},
  url = 	 {https://proceedings.mlr.press/v75/belkin18a.html},
  abstract = 	 {Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data.  In this paper we take the approximation theory point of view to explore various aspects of  smooth kernels related to their inferential properties. We  analyze eigenvalue decay of  kernels operators and  matrices,  properties of eigenfunctions/eigenvectors and “Fourier” coefficients of functions in the kernel space restricted to a discrete set of data points. We also investigate the fitting capacity of kernels,  giving explicit bounds on the fat shattering dimension of the balls in  Reproducing Kernel Hilbert spaces.  Interestingly, the same properties that make kernels  very effective approximators for functions in their “native” kernel space,  also limit their capacity to represent arbitrary functions.  We discuss various implications, including those for gradient descent type methods. It is important to note that most of our  bounds are measure independent.  Moreover,  at least in moderate dimension, the bounds for eigenvalues are much tighter than the bounds which can be obtained from the usual matrix concentration results. For example, we see that  eigenvalues of kernel matrices show nearly exponential decay with constants depending only on the kernel and the domain. We call this “approximation beats concentration” phenomenon as even when the data are sampled from a probability distribution, some of their aspects are better understood in terms of approximation theory.  }
}


@inproceedings{rudner2023FuncReg,
  title     = {Function-Space Regularization in Neural Networks: A Probabilistic Perspective},
  author    = {Rudner, Tim G. J. and Kapoor, Sanyam and Qiu, Shikai and Wilson, Andrew Gordon},
  booktitle = {Proceedings of the 40th International Conference on Machine Learning},
  pages     = {29275--29290},
  year      = {2023},
  editor    = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan},
  volume    = {202},
  series    = {Proceedings of Machine Learning Research},
  month     = {23--29 Jul},
  publisher = {PMLR},
  pdf       = {https://proceedings.mlr.press/v202/rudner23a/rudner23a.pdf},
  url       = {https://proceedings.mlr.press/v202/rudner23a.html},
  abstract  = {Parameter-space regularization in neural network optimization is a fundamental tool for improving generalization. However, standard parameter-space regularization methods make it challenging to encode explicit preferences about desired predictive functions into neural network training. In this work, we approach regularization in neural networks from a probabilistic perspective and show that by viewing parameter-space regularization as specifying an empirical prior distribution over the model parameters, we can derive a probabilistically well-motivated regularization technique that allows explicitly encoding information about desired predictive functions into neural network training. This method—which we refer to as function-space empirical Bayes (FS-EB)—includes both parameter- and function-space regularization, is mathematically simple, easy to implement, and incurs only minimal computational overhead compared to standard regularization techniques. We evaluate the utility of this regularization technique empirically and demonstrate that the proposed method leads to near-perfect semantic shift detection, highly-calibrated predictive uncertainty estimates, successful task adaption from pre-trained models, and improved generalization under covariate shift.}
}


@inproceedings{cinquin2021pathologies,
  title     = {Pathologies in Priors and Inference for Bayesian Transformers},
  author    = {Tristan Cinquin and Alexander Immer and Max Horn and Vincent Fortuin},
  booktitle = {I (Still) Can't Believe It's Not Better! NeurIPS 2021 Workshop},
  year      = {2021},
  url       = {https://openreview.net/forum?id=Gv41ucDhhlY}
}

@InProceedings{blundell2015weight,
  title = 	 {Weight Uncertainty in Neural Network},
  author = 	 {Blundell, Charles and Cornebise, Julien and Kavukcuoglu, Koray and Wierstra, Daan},
  booktitle = 	 {Proceedings of the 32nd International Conference on Machine Learning},
  pages = 	 {1613--1622},
  year = 	 {2015},
  editor = 	 {Bach, Francis and Blei, David},
  volume = 	 {37},
  series = 	 {Proceedings of Machine Learning Research},
  address = 	 {Lille, France},
  month = 	 {07--09 Jul},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v37/blundell15.pdf},
  url = 	 {https://proceedings.mlr.press/v37/blundell15.html},
  abstract = 	 {We introduce a new, efficient, principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. It regularises the weights by minimising a compression cost, known as the variational free energy or the expected lower bound on the marginal likelihood. We show that this principled kind of regularisation yields comparable performance to dropout on MNIST classification. We then demonstrate how the learnt uncertainty in the weights can be used to improve generalisation in non-linear regression problems, and how this weight uncertainty can be used to drive the exploration-exploitation trade-off in reinforcement learning.}
}

@inproceedings{wu2023indirect,
  title     = {Indirect Functional Bayesian Neural Networks},
  author    = {Mengjing Wu and Junyu Xuan and Jie Lu},
  booktitle = {Fifth Symposium on Advances in Approximate Bayesian Inference},
  year      = {2023},
  url       = {https://openreview.net/forum?id=-xfwSaifkU}
}

@inproceedings{hensman2013gaussian,
author = {Hensman, James and Fusi, Nicol\`{o} and Lawrence, Neil D.},
title = {Gaussian processes for Big data},
year = {2013},
publisher = {AUAI Press},
address = {Arlington, Virginia, USA},
abstract = {We introduce stochastic variational inference for Gaussian process models. This enables the application of Gaussian process (GP) models to data sets containing millions of data points. We show how GPs can be variationally decomposed to depend on a set of globally relevant inducing variables which factorize the model in the necessary manner to perform variational inference. Our approach is readily extended to models with non-Gaussian likelihoods and latent variable models based around Gaussian processes. We demonstrate the approach on a simple toy problem and two real world data sets.},
booktitle = {Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence},
pages = {282–290},
numpages = {9},
location = {Bellevue, WA},
series = {UAI'13}
}

@inproceedings{malinin2021uncertGBM,
    title={Uncertainty in Gradient Boosting via Ensembles},
    author={Andrey Malinin and Liudmila Prokhorenkova and Aleksei Ustimenko},
    booktitle={International Conference on Learning Representations},
    year={2021},
    url={https://openreview.net/forum?id=1Jv6b0Zq3qi}
}

@misc{Dua2019UCI, 
  author = {Dua, Dheeru and Graff, Casey}, 
  year = {2017}, 
  title = {UCI Machine Learning Repository}, 
  url = {http://archive.ics.uci.edu/ml}, 
  institution = {University of California, Irvine, School of Information and Computer Sciences} 
}

@inproceedings{khan2020approximate,
 author = {Khan, Mohammad Emtiyaz and Immer, Alexander and Abedi, Ehsan and Korzepa, Maciej},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Approximate Inference Turns Deep Networks into Gaussian Processes},
 url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/b3bbccd6c008e727785cb81b1aa08ac5-Paper.pdf},
 volume = {32},
 year = {2019}
}

@article{hoffman13SVI,
  author  = {Matthew D. Hoffman and David M. Blei and Chong Wang and John Paisley},
  title   = {Stochastic Variational Inference},
  journal = {Journal of Machine Learning Research},
  year    = {2013},
  volume  = {14},
  number  = {40},
  pages   = {1303--1347},
  url     = {http://jmlr.org/papers/v14/hoffman13a.html}
}

@inproceedings{tomczak2020lowrankGausVI,
 author = {Tomczak, Marcin and Swaroop, Siddharth and Turner, Richard},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Larochelle and M. Ranzato and R. Hadsell and M.F. Balcan and H. Lin},
 pages = {4610--4622},
 publisher = {Curran Associates, Inc.},
 title = {Efficient Low Rank Gaussian Variational Inference for Neural Networks},
 url = {https://proceedings.neurips.cc/paper_files/paper/2020/file/310cc7ca5a76a446f85c1a0d641ba96d-Paper.pdf},
 volume = {33},
 year = {2020}
}

@InProceedings{wilson2022evalBNN,
  title = 	 {Evaluating Approximate Inference in Bayesian Deep Learning},
  author =       {Wilson, Andrew Gordon and Izmailov, Pavel and Hoffman, Matthew D and Gal, Yarin and Li, Yingzhen and Pradier, Melanie F and Vikram, Sharad and Foong, Andrew and Lotfi, Sanae and Farquhar, Sebastian},
  booktitle = 	 {Proceedings of the NeurIPS 2021 Competitions and Demonstrations Track},
  pages = 	 {113--124},
  year = 	 {2022},
  editor = 	 {Kiela, Douwe and Ciccone, Marco and Caputo, Barbara},
  volume = 	 {176},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {06--14 Dec},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v176/wilson22a/wilson22a.pdf},
  url = 	 {https://proceedings.mlr.press/v176/wilson22a.html},
  abstract = 	 {Uncertainty representation is crucial to the safe and reliable deployment of deep learning. Bayesian methods provide a natural mechanism to represent epistemic uncertainty, leading to improved generalization and calibrated predictive distributions. Understanding the fidelity of approximate inference has extraordinary value beyond the standard approach of measuring generalization on a particular task: if approximate inference is working correctly, then we can expect more reliable and accurate deployment across any number of real-world settings. In this competition, we evaluate the fidelity of approximate Bayesian inference procedures in deep learning, using as a reference Hamiltonian Monte Carlo (HMC) samples obtained by parallelizing computations over hundreds of tensor processing unit (TPU) devices. We consider a variety of tasks, including image recognition, regression, covariate shift, and medical applications. All data are publicly available, and we release several baselines, including stochastic MCMC, variational methods, and deep ensembles. The competition resulted in hundreds of submissions across many teams. The winning entries all involved novel multi-modal posterior approximations, highlighting the relative importance of representing multiple modes, and suggesting that we should not consider deep ensembles a {“}non-Bayesian{”} alternative to standard unimodal approximations. In the future, the competition will provide a foundation for innovation and continued benchmarking of approximate Bayesian inference procedures in deep learning. The HMC samples will remain available through the competition website.}
}

@inproceedings{kingma2022autoencoding,
  abstract = {How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions is two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, we show that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. Theoretical advantages are reflected in experimental results.},
  added-at = {2020-10-15T14:36:56.000+0200},
  author = {Kingma, Diederik P. and Welling, Max},
  biburl = {https://www.bibsonomy.org/bibtex/242e5be6faa01cba2587f4907ac99dce8/annakrause},
  booktitle = {2nd International Conference on Learning Representations, {ICLR} 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings},
  eprint = {http://arxiv.org/abs/1312.6114v10},
  eprintclass = {stat.ML},
  eprinttype = {arXiv},
  file = {:http\://arxiv.org/pdf/1312.6114v10:PDF;:KingmaWelling_Auto-EncodingVariationalBayes.pdf:PDF},
  interhash = {a626a9d77a123c52405a08da983203cb},
  intrahash = {42e5be6faa01cba2587f4907ac99dce8},
  keywords = {cs.LG stat.ML vae},
  timestamp = {2021-02-01T17:13:18.000+0100},
  title = {{Auto-Encoding Variational Bayes}},
  year = 2014
}

@misc{wandb,
    title = {Experiment Tracking with Weights and Biases},
    year = {2020},
    note = {Software available from wandb.com},
    url={https://www.wandb.com/},
    author = {Biewald, Lukas},
}

@misc{kingma2017adam,
    title={Adam: A Method for Stochastic Optimization}, 
    author={Diederik P. Kingma and Jimmy Ba},
    year={2017},
    eprint={1412.6980},
    archivePrefix={arXiv},
    primaryClass={cs.LG}
}

@misc{jax2018github,
    author = {James Bradbury and Roy Frostig and Peter Hawkins and Matthew James Johnson and Chris Leary and Dougal Maclaurin and George Necula and Adam Paszke and Jake Vander{P}las and Skye Wanderman-{M}ilne and Qiao Zhang},
    title = {{JAX}: composable transformations of {P}ython+{N}um{P}y programs},
    url = {http://github.com/google/jax},
    version = {0.3.13},
    year = {2018},
}

@misc{haiku2020github,
  author = {Tom Hennigan and Trevor Cai and Tamara Norman and Lena Martens and Igor Babuschkin},
  title = {{H}aiku: {S}onnet for {JAX}},
  url = {http://github.com/deepmind/dm-haiku},
  version = {0.0.10},
  year = {2020},
}

@article{Pinder2022gpjax,
  doi = {10.21105/joss.04455},
  url = {https://doi.org/10.21105/joss.04455},
  year = {2022},
  publisher = {The Open Journal},
  volume = {7},
  number = {75},
  pages = {4455},
  author = {Thomas Pinder and Daniel Dodd},
  title = {GPJax: A Gaussian Process Framework in JAX},
  journal = {Journal of Open Source Software}
}

@article{flamary2021pot,
  author  = {R{\'e}mi Flamary and Nicolas Courty and Alexandre Gramfort and Mokhtar Z. Alaya and Aur{\'e}lie Boisbunon and Stanislas Chambon and Laetitia Chapel and Adrien Corenflos and Kilian Fatras and Nemo Fournier and L{\'e}o Gautheron and Nathalie T.H. Gayraud and Hicham Janati and Alain Rakotomamonjy and Ievgen Redko and Antoine Rolet and Antony Schutz and Vivien Seguy and Danica J. Sutherland and Romain Tavenard and Alexander Tong and Titouan Vayer},
  title   = {POT: Python Optimal Transport},
  journal = {Journal of Machine Learning Research},
  year    = {2021},
  volume  = {22},
  number  = {78},
  pages   = {1-8},
  url     = {http://jmlr.org/papers/v22/20-451.html}
}

@phdthesis{wong2023MLinFin,
	author = {Wong, Yuk Kwan},
	title = {Machine learning in portfolio management},
	url = {https://hdl.handle.net/2123/31701},
	pages = {},
	year = {2023},
	school = {University of Sydney}
}

@article{medicineBDL,
    author = {Abdullah, Abdullah and Hassan, Masoud and Mustafa, Yaseen},
    year = {2022},
    month = {01},
    pages = {1-1},
    title = {A Review on {B}ayesian Deep Learning in Healthcare: Applications and Challenges},
    volume = {10},
    journal = {IEEE Access},
    doi = {10.1109/ACCESS.2022.3163384}
}

@article{uncertaintymedicalML,
    author = {Kompa, Benjamin and Snoek, Jasper and Beam, Andrew},
    year = {2021},
    month = {12},
    pages = {},
    title = {Second opinion needed: communicating uncertainty in medical machine learning},
    volume = {4},
    journal = {npj Digital Medicine},
    doi = {10.1038/s41746-020-00367-3}
}


@ARTICLE{Krishnapriya2023MRIBdL,
    AUTHOR={Krishnapriya, Srigiri and Karuna, Yepuganti},   
	TITLE={Pre-trained deep learning models for brain MRI image classification},      
	JOURNAL={Frontiers in Human Neuroscience},      
    VOLUME={17},           
    YEAR={2023},      
    URL={https://www.frontiersin.org/articles/10.3389/fnhum.2023.1150120},       
    DOI={10.3389/fnhum.2023.1150120},      
    ISSN={1662-5161},   
    ABSTRACT={Brain tumors are serious conditions caused by uncontrolled and abnormal cell division. Tumors can have devastating implications if not accurately and promptly detected. Magnetic resonance imaging (MRI) is one of the methods frequently used to detect brain tumors owing to its excellent resolution. In the past few decades, substantial research has been conducted in the field of classifying brain images, ranging from traditional methods to deep-learning techniques such as convolutional neural networks (CNN). To accomplish classification, machine-learning methods require manually created features. In contrast, CNN achieves classification by extracting visual features from unprocessed images. The size of the training dataset had a significant impact on the features that CNN extracts. The CNN tends to overfit when its size is small. Deep CNNs (DCNN) with transfer learning have therefore been developed. The aim of this work was to investigate the brain MR image categorization potential of pre-trained DCNN VGG-19, VGG-16, ResNet50, and Inception V3 models using data augmentation and transfer learning techniques. Validation of the test set utilizing accuracy, recall, Precision, and F1 score showed that the pre-trained VGG-19 model with transfer learning exhibited the best performance. In addition, these methods offer an end-to-end classification of raw images without the need for manual attribute extraction.}
}


@article {analystRecommendationBML,
	author = {Bew, David and Harvey, Campbell R. and Ledford, Anthony and Radnor, Sam and Sinclair, Andrew},
	title = {Modeling Analysts{\textquoteright} Recommendations via {B}ayesian Machine Learning},
	volume = {1},
	number = {1},
	pages = {75--98},
	year = {2019},
	doi = {10.3905/jfds.2019.1.1.075},
	publisher = {Institutional Investor Journals Umbrella},
	issn = {2640-3943},
	URL = {https://jfds.pm-research.com/content/1/1/75},
	eprint = {https://jfds.pm-research.com/content/1/1/75.full.pdf},
	journal = {The Journal of Financial Data Science}
}

@INPROCEEDINGS{Bertin-Mahieux2011,
  author = {Thierry Bertin-Mahieux and Daniel P.W. Ellis and Brian Whitman and Paul Lamere},
  title = {The Million Song Dataset},
  booktitle = {{Proceedings of the 12th International Conference on Music Information
	Retrieval ({ISMIR} 2011)}},
  year = {2011},
  owner = {thierry},
  timestamp = {2010.03.07}
}

@inproceedings{milios2018DirGP,
    author = {Milios, Dimitrios and Camoriano, Raffaello and Michiardi, Pietro and Rosasco, Lorenzo and Filippone, Maurizio},
    booktitle = {Advances in Neural Information Processing Systems},
    editor = {S. Bengio and H. Wallach and H. Larochelle and K. Grauman and N. Cesa-Bianchi and R. Garnett},
    pages = {},
    publisher = {Curran Associates, Inc.},
    title = {Dirichlet-based Gaussian Processes for Large-scale Calibrated Classification},
    url = {https://proceedings.neurips.cc/paper_files/paper/2018/file/b6617980ce90f637e68c3ebe8b9be745-Paper.pdf},
    volume = {31},
    year = {2018}
}


@article{lecun2010mnist,
  title={MNIST handwritten digit database},
  author={LeCun, Yann and Cortes, Corinna and Burges, CJ},
  journal={ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist},
  volume={2},
  year={2010}
}

@misc{xiao2017FMNIST,
  author       = {Han Xiao and Kashif Rasul and Roland Vollgraf},
  title        = {Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms},
  date         = {2017-08-28},
  year         = {2017},
  eprintclass  = {cs.LG},
  eprinttype   = {arXiv},
  eprint       = {cs.LG/1708.07747},
}

@inproceedings{osawa2019practical,
 author = {Osawa, Kazuki and Swaroop, Siddharth and Khan, Mohammad Emtiyaz and Jain, Anirudh and Eschenhagen, Runa and Turner, Richard E and Yokota, Rio},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Practical Deep Learning with Bayesian Principles},
 url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/b53477c2821c1bf0da5d40e57b870d35-Paper.pdf},
 volume = {32},
 year = {2019}
}

@article{scikitlearn,
  title={Scikit-learn: Machine Learning in {P}ython},
  author={Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V.
          and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P.
          and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and
          Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.},
  journal={Journal of Machine Learning Research},
  volume={12},
  pages={2825--2830},
  year={2011}
}

@inproceedings{sensoy2018evidential,
 author = {Sensoy, Murat and Kaplan, Lance and Kandemir, Melih},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {S. Bengio and H. Wallach and H. Larochelle and K. Grauman and N. Cesa-Bianchi and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Evidential Deep Learning to Quantify Classification Uncertainty},
 url = {https://proceedings.neurips.cc/paper_files/paper/2018/file/a981f2b708044d6fb4a71a1463242520-Paper.pdf},
 volume = {31},
 year = {2018}
}

@misc{clanuwat2018kmnist,
author = {Clanuwat, Tarin and Bober-Irizar, Mikel and Kitamoto, Asanobu and Lamb, Alex and Kazuaki, Yamamoto and Ha, David},
year = {2018},
month = {12},
pages = {},
title = {Deep Learning for Classical Japanese Literature},
doi = {10.20676/00000341}
}

@inproceedings{liu2019svgd,
 author = {Liu, Qiang and Wang, Dilin},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {D. Lee and M. Sugiyama and U. Luxburg and I. Guyon and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm},
 url = {https://proceedings.neurips.cc/paper_files/paper/2016/file/b3ba8f1bee1238a2f37603d90b58898d-Paper.pdf},
 volume = {29},
 year = {2016}
}

@dataset{lilly2021GulfDrifters,
author       = {Lilly, Jonathan M. and
          Pérez-Brunius, Paula},
title        = {{GulfDrifters: A consolidated surface drifter 
           dataset for the Gulf of Mexico}},
month        = jan,
year         = 2021,
publisher    = {Zenodo},
version      = {1.1.0},
doi          = {10.5281/zenodo.4421585},
url          = {https://doi.org/10.5281/zenodo.4421585}
}

@misc{shalashilin2024GPocean,
author = {Ivan Shalashilin},
title =  {Gaussian Processes for Vector Fields and Ocean Current Modelling},
month = Mar,
year = 2024,
url= {https://docs.jaxgaussianprocesses.com/examples/oceanmodelling/}
}

@InProceedings{berlinghieri2023gaussian,
  title = 	 {{G}aussian processes at the Helm(holtz): A more fluid model for ocean currents},
  author =       {Berlinghieri, Renato and Trippe, Brian L. and Burt, David R. and Giordano, Ryan James and Srinivasan, Kaushik and \"{O}zg\"{o}kmen, Tamay and Xia, Junfei and Broderick, Tamara},
  booktitle = 	 {Proceedings of the 40th International Conference on Machine Learning},
  pages = 	 {2113--2163},
  year = 	 {2023},
  editor = 	 {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan},
  volume = 	 {202},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {23--29 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v202/berlinghieri23a/berlinghieri23a.pdf},
  url = 	 {https://proceedings.mlr.press/v202/berlinghieri23a.html},
  abstract = 	 {Oceanographers are interested in predicting ocean currents and identifying divergences in a current vector field based on sparse observations of buoy velocities. Since we expect current dynamics to be smooth but highly non-linear, Gaussian processes (GPs) offer an attractive model. But we show that applying a GP with a standard stationary kernel directly to buoy data can struggle at both current prediction and divergence identification – due to some physically unrealistic prior assumptions. To better reflect known physical properties of currents, we propose to instead put a standard stationary kernel on the divergence and curl-free components of a vector field obtained through a Helmholtz decomposition. We show that, because this decomposition relates to the original vector field just via mixed partial derivatives, we can still perform inference given the original data with only a small constant multiple of additional computational expense. We illustrate the benefits of our method on synthetic and real oceans data.}
}

@inproceedings{cinquin2024fsplaplace,
    title={{FSP}-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning},
    author={Tristan Cinquin and Marvin Pf{\"o}rtner and Vincent Fortuin and Philipp Hennig and Robert Bamler},
    booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
    year={2024},
    url={https://openreview.net/forum?id=83vxe8alV4}
}

@misc{quang2017infinitedimensionallogdeterminantdivergencesii,
      title={Infinite-dimensional Log-Determinant divergences II: Alpha-Beta divergences}, 
      author={Minh Ha Quang},
      year={2017},
      eprint={1610.08087},
      archivePrefix={arXiv},
      primaryClass={math.FA},
      url={https://arxiv.org/abs/1610.08087}, 
}

@misc{sam2024bayesianneuralnetworksdomain,
      title={Bayesian Neural Networks with Domain Knowledge Priors}, 
      author={Dylan Sam and Rattana Pukdee and Daniel P. Jeong and Yewon Byun and J. Zico Kolter},
      year={2024},
      eprint={2402.13410},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2402.13410}, 
}

@inproceedings{yang2020outputconstrainedBNN,
 author = {Yang, Wanqian and Lorch, Lars and Graule, Moritz and Lakkaraju, Himabindu and Doshi-Velez, Finale},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Larochelle and M. Ranzato and R. Hadsell and M.F. Balcan and H. Lin},
 pages = {12721--12731},
 publisher = {Curran Associates, Inc.},
 title = {Incorporating Interpretable Output Constraints in Bayesian Neural Networks},
 url = {https://proceedings.neurips.cc/paper_files/paper/2020/file/95c7dfc5538e1ce71301cf92a9a96bd0-Paper.pdf},
 volume = {33},
 year = {2020}
}
