% Review of UQ for deep learning
@article{abdar2021review,
  title={A review of uncertainty quantification in deep learning: Techniques, applications and challenges},
  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 others},
  journal={Information Fusion},
  volume={76},
  pages={243--297},
  year={2021},
  publisher={Elsevier}
}
% Medical decision-making needs UQ
@article{begoli2019need,
  title={The need for uncertainty quantification in machine-assisted medical decision making},
  author={Begoli, Edmon and Bhattacharya, Tanmoy and Kusnezov, Dimitri},
  journal={Nature Machine Intelligence},
  volume={1},
  number={1},
  pages={20--23},
  year={2019},
  publisher={Nature Publishing Group}
}
% Self-driving cars
@article{michelmore2018evaluating,
  title={Evaluating uncertainty quantification in end-to-end autonomous driving control},
  author={Michelmore, Rhiannon and Kwiatkowska, Marta and Gal, Yarin},
  journal={arXiv preprint arXiv:1811.06817},
  year={2018}
}
% Exhaustive review of UQ in deep learning
@article{gawlikowski2021survey,
  title={A survey of uncertainty in deep neural networks},
  author={Gawlikowski, Jakob and Tassi, Cedrique Rovile Njieutcheu and Ali, Mohsin and Lee, Jongseok and Humt, Matthias and Feng, Jianxiang and Kruspe, Anna and Triebel, Rudolph and Jung, Peter and Roscher, Ribana and others},
  journal={arXiv preprint arXiv:2107.03342},
  year={2021}
}
% Erroneous predictions of DL systems
@article{ovadia2019can,
  title={Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift},
  author={Ovadia, Yaniv and Fertig, Emily and Ren, Jie and Nado, Zachary and Sculley, David and Nowozin, Sebastian and Dillon, Joshua and Lakshminarayanan, Balaji and Snoek, Jasper},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}
% Uncertainties in CV
@article{kendall2017uncertainties,
  title={What uncertainties do we need in bayesian deep learning for computer vision?},
  author={Kendall, Alex and Gal, Yarin},
  journal={Advances in neural information processing systems},
  volume={30},
  year={2017}
}
% Disentangling UQ
@inproceedings{valdenegro2022deeper,
  title={A deeper look into aleatoric and epistemic uncertainty disentanglement},
  author={Valdenegro-Toro, Matias and Mori, Daniel Saromo},
  booktitle={2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)},
  pages={1508--1516},
  year={2022},
  organization={IEEE}
}
% Another review of UQ in deep learning
@article{hullermeier2019aleatoric,
  title={Aleatoric and epistemic uncertainty in machine learning: A tutorial introduction},
  author={H{\"u}llermeier, Eyke and Waegeman, Willem},
  year={2019}
}
% The case for BDL
@article{wilson2020case,
  title={The case for Bayesian deep learning},
  author={Wilson, Andrew Gordon},
  journal={arXiv preprint arXiv:2001.10995},
  year={2020}
}
% Ensembles approximate a BMA
@article{wilson2020bayesian,
  title={Bayesian deep learning and a probabilistic perspective of generalization},
  author={Wilson, Andrew G and Izmailov, Pavel},
  journal={Advances in neural information processing systems},
  volume={33},
  pages={4697--4708},
  year={2020}
}

%DE
@article{lakshminarayanan2017de,
  title={Simple and scalable predictive uncertainty estimation using deep ensembles},
  author={Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles},
  journal={Advances in neural information processing systems},
  volume={30},
  year={2017}
}


@article{unni2020deep,
  title={Deep convolutional mixture density network for inverse design of layered photonic structures},
  author={Unni, Rohit and Yao, Kan and Zheng, Yuebing},
  journal={ACS Photonics},
  volume={7},
  number={10},
  pages={2703--2712},
  year={2020},
  publisher={ACS Publications}
}


@article{d2018photometric,
  title={Photometric redshift estimation via deep learning-generalized and pre-classification-less, image based, fully probabilistic redshifts},
  author={D’Isanto, Antonio and Polsterer, Kai Lars},
  journal={Astronomy \& Astrophysics},
  volume={609},
  pages={A111},
  year={2018},
  publisher={EDP Sciences}
}


@article{davis2020use,
  title={The use of mixture density networks in the emulation of complex epidemiological individual-based models},
  author={Davis, Christopher N and Hollingsworth, T Deirdre and Caudron, Quentin and Irvine, Michael A},
  journal={PLoS computational biology},
  volume={16},
  number={3},
  pages={e1006869},
  year={2020},
  publisher={Public Library of Science San Francisco, CA USA}
}


@article{zhang2020improved,
  title={Improved deep mixture density network for regional wind power probabilistic forecasting},
  author={Zhang, Hao and Liu, Yongqian and Yan, Jie and Han, Shuang and Li, Li and Long, Quan},
  journal={IEEE Transactions on Power Systems},
  volume={35},
  number={4},
  pages={2549--2560},
  year={2020},
  publisher={IEEE}
}


@inproceedings{zen2014deep,
  title={Deep mixture density networks for acoustic modeling in statistical parametric speech synthesis},
  author={Zen, Heiga and Senior, Andrew},
  booktitle={2014 IEEE international conference on acoustics, speech and signal processing (ICASSP)},
  pages={3844--3848},
  year={2014},
  organization={IEEE}
}


@article{bazzani2016recurrent,
  title={Recurrent mixture density network for spatiotemporal visual attention},
  author={Bazzani, Loris and Larochelle, Hugo and Torresani, Lorenzo},
  journal={arXiv preprint arXiv:1603.08199},
  year={2016}
}


@article{bishop1994mixture,
  title={Mixture density networks},
  author={Bishop, Christopher M},
  year={1994},
  publisher={Aston University}
}

@inproceedings{chen2014stochastic,
  title={Stochastic gradient Hamiltonian Monte Carlo},
  author={Chen, Tianqi and Fox, Emily and Guestrin, Carlos},
  booktitle={International conference on machine learning},
  pages={1683--1691},
  year={2014},
  organization={PMLR}
}

@article{daxberger2021laplace,
  title={Laplace redux-effortless bayesian deep learning},
  author={Daxberger, Erik and Kristiadi, Agustinus and Immer, Alexander and Eschenhagen, Runa and Bauer, Matthias and Hennig, Philipp},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={20089--20103},
  year={2021}
}

@article{hoffman2013stochastic,
  title={Stochastic variational inference},
  author={Hoffman, Matthew D and Blei, David M and Wang, Chong and Paisley, John},
  journal={Journal of Machine Learning Research},
  year={2013}
}

@article{graves2011practical,
  title={Practical variational inference for neural networks},
  author={Graves, Alex},
  journal={Advances in neural information processing systems},
  volume={24},
  year={2011}
}

@book{neal2012bayesian,
  title={Bayesian learning for neural networks},
  author={Neal, Radford M},
  volume={118},
  year={2012},
  publisher={Springer Science \& Business Media}
}

@article{bacharoglou2010approximation,
  title={Approximation of probability distributions by convex mixtures of Gaussian measures},
  author={Bacharoglou, Athanassia},
  journal={Proceedings of the American Mathematical Society},
  volume={138},
  number={7},
  pages={2619--2628},
  year={2010}
}

@book{goodfellow2016deep,
  title={Deep learning},
  author={Goodfellow, Ian and Bengio, Yoshua and Courville, Aaron},
  year={2016},
  publisher={MIT press}
}

% PBP
@inproceedings{hernandez2015probabilistic,
  title={Probabilistic backpropagation for scalable learning of bayesian neural networks},
  author={Hern{\'a}ndez-Lobato, Jos{\'e} Miguel and Adams, Ryan},
  booktitle={International conference on machine learning},
  pages={1861--1869},
  year={2015},
  organization={PMLR}
}

% MCD
@inproceedings{gal2016dropout,
  title={Dropout as a bayesian approximation: Representing model uncertainty in deep learning},
  author={Gal, Yarin and Ghahramani, Zoubin},
  booktitle={international conference on machine learning},
  pages={1050--1059},
  year={2016},
  organization={PMLR}
}

% MCBN
@inproceedings{teye2018bayesian,
  title={Bayesian uncertainty estimation for batch normalized deep networks},
  author={Teye, Mattias and Azizpour, Hossein and Smith, Kevin},
  booktitle={International Conference on Machine Learning},
  pages={4907--4916},
  year={2018},
  organization={PMLR}
}

% DE
@article{lakshminarayanan2017simple,
  title={Simple and scalable predictive uncertainty estimation using deep ensembles},
  author={Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles},
  journal={Advances in neural information processing systems},
  volume={30},
  year={2017}
}


@book{leoni2017first,
  title={A first course in Sobolev spaces},
  author={Leoni, Giovanni},
  year={2017},
  publisher={American Mathematical Soc.}
}

% Minka paper on EM
@article{minka1998expectation,
  title={Expectation-Maximization as lower bound maximization},
  author={Minka, Thomas},
  journal={Tutorial published on the web at http://www-white. media. mit. edu/tpminka/papers/em. html},
  volume={7},
  number={2},
  year={1998}
}

% uncertainty as an acquisition function
@inproceedings{lewis1995sequential,
  title={A sequential algorithm for training text classifiers: Corrigendum and additional data},
  author={Lewis, David D},
  booktitle={Acm Sigir Forum},
  volume={29},
  number={2},
  pages={13--19},
  year={1995},
  organization={ACM New York, NY, USA}
}

% deep RL uncertyainty
@article{clements2019estimating,
  title={Estimating risk and uncertainty in deep reinforcement learning},
  author={Clements, William R and Van Delft, Bastien and Robaglia, Beno{\^\i}t-Marie and Slaoui, Reda Bahi and Toth, S{\'e}bastien},
  journal={arXiv preprint arXiv:1905.09638},
  year={2019}
}
@article{charpentier2022disentangling,
  title={Disentangling Epistemic and Aleatoric Uncertainty in Reinforcement Learning},
  author={Charpentier, Bertrand and Senanayake, Ransalu and Kochenderfer, Mykel and G{\"u}nnemann, Stephan},
  journal={arXiv preprint arXiv:2206.01558},
  year={2022}
}


@article{farrell2021deep,
  title={Deep neural networks for estimation and inference},
  author={Farrell, Max H and Liang, Tengyuan and Misra, Sanjog},
  journal={Econometrica},
  volume={89},
  number={1},
  pages={181--213},
  year={2021},
  publisher={Wiley Online Library}
}


@article{wu1983convergence,
  title={On the convergence properties of the EM algorithm},
  author={Wu, CF Jeff},
  journal={The Annals of statistics},
  pages={95--103},
  year={1983},
  publisher={JSTOR}
}


@inproceedings{arora2019fine,
  title={Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks},
  author={Arora, Sanjeev and Du, Simon and Hu, Wei and Li, Zhiyuan and Wang, Ruosong},
  booktitle={International Conference on Machine Learning},
  pages={322--332},
  year={2019},
  organization={PMLR}
}

% Adversarial examples
@article{goodfellow2014explaining,
  title={Explaining and harnessing adversarial examples},
  author={Goodfellow, Ian J and Shlens, Jonathon and Szegedy, Christian},
  journal={arXiv preprint arXiv:1412.6572},
  year={2014}
}


% DE, Bayesian perspective
@article{hoffmann2021deep,
  title={Deep ensembles from a Bayesian perspective},
  author={Hoffmann, Lara and Elster, Clemens},
  journal={arXiv preprint arXiv:2105.13283},
  year={2021}
}

% Deep split ensembles
@article{sarawgi2020have,
  title={Why have a unified predictive uncertainty? disentangling it using deep split ensembles},
  author={Sarawgi, Utkarsh and Zulfikar, Wazeer and Khincha, Rishab and Maes, Pattie},
  journal={arXiv preprint arXiv:2009.12406},
  year={2020}
}
% Anchored deep ensembles
@inproceedings{pearce2020uncertainty,
  title={Uncertainty in neural networks: Approximately bayesian ensembling},
  author={Pearce, Tim and Leibfried, Felix and Brintrup, Alexandra},
  booktitle={International conference on artificial intelligence and statistics},
  pages={234--244},
  year={2020},
  organization={PMLR}
}

% UCI repository
@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" }

% Power Plant dataset
@article{tufekci2014prediction,
  title={Prediction of full load electrical power output of a base load operated combined cycle power plant using machine learning methods},
  author={T{\"u}fekci, P{\i}nar},
  journal={International Journal of Electrical Power \& Energy Systems},
  volume={60},
  pages={126--140},
  year={2014},
  publisher={Elsevier}
}

%Concrete dataset neural network
@article{yeh1998modeling,
  title={Modeling of strength of high-performance concrete using artificial neural networks},
  author={Yeh, I-C},
  journal={Cement and Concrete research},
  volume={28},
  number={12},
  pages={1797--1808},
  year={1998},
  publisher={Elsevier}
}

%Energy dataset
@article{tsanas2012accurate,
  title={Accurate quantitative estimation of energy performance of residential buildings using statistical machine learning tools},
  author={Tsanas, Athanasios and Xifara, Angeliki},
  journal={Energy and buildings},
  volume={49},
  pages={560--567},
  year={2012},
  publisher={Elsevier}
}

% Laplace bridge
@inproceedings{hobbhahn2022fast,
  title={Fast predictive uncertainty for classification with bayesian deep networks},
  author={Hobbhahn, Marius and Kristiadi, Agustinus and Hennig, Philipp},
  booktitle={Uncertainty in Artificial Intelligence},
  pages={822--832},
  year={2022},
  organization={PMLR}
}

% LSTM
@article{LSTMhochreiter,
author = {Hochreiter, Sepp and Schmidhuber, Jürgen},
year = {1997},
month = {12},
pages = {1735-80},
title = {Long Short-term Memory},
volume = {9},
journal = {Neural computation},
}

%Dist shift
@inproceedings{masserano2022adaptive,
title={Adaptive Sampling for Probabilistic Forecasting under Distribution Shift},
author={Luca Masserano and Syama Sundar Rangapuram and Shubham Kapoor and Rajbir Singh Nirwan and Youngsuk Park and Michael Bohlke-Schneider},
booktitle={NeurIPS 2022 Workshop on Distribution Shifts: Connecting Methods and Applications},
year={2022},
url={https://openreview.net/forum?id=oPHuNpJl3c}
}

% MultiSWAG
@inproceedings{NEURIPS2020MultiSWAG,
 author = {Wilson, Andrew G and Izmailov, Pavel},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Larochelle and M. Ranzato and R. Hadsell and M.F. Balcan and H. Lin},
 pages = {4697--4708},
 publisher = {Curran Associates, Inc.},
 title = {Bayesian Deep Learning and a Probabilistic Perspective of Generalization},
 volume = {33},
 year = {2020}
}

% Neural EM
@inproceedings{NIPS2017_NEM,
 author = {Greff, Klaus and van Steenkiste, Sjoerd and Schmidhuber, J\"{u}rgen},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {I. Guyon and U. Von Luxburg and S. Bengio and H. Wallach and R. Fergus and S. Vishwanathan and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Neural Expectation Maximization},
 volume = {30},
 year = {2017}
}

% stock forecasting covid-19
@article{chandra2021bayesian,
  title={Bayesian neural networks for stock price forecasting before and during COVID-19 pandemic},
  author={Chandra, Rohitash and He, Yixuan},
  journal={Plos one},
  volume={16},
  number={7},
  pages={e0253217},
  year={2021},
  publisher={Public Library of Science San Francisco, CA USA}
}

% Weight initialization
@inproceedings{glorot2010understanding,
  title={Understanding the difficulty of training deep feedforward neural networks},
  author={Glorot, Xavier and Bengio, Yoshua},
  booktitle={Proceedings of the thirteenth international conference on artificial intelligence and statistics},
  pages={249--256},
  year={2010},
  organization={JMLR Workshop and Conference Proceedings}
}


