### Introduction

Machine learning models have traditionally focused on point predictions, aiming to provide accurate estimates of target variables given input features. However, in many real-world applications, understanding the uncertainty associated with these predictions is equally important. Consequently, there has been a growing interest in methods that move beyond point estimates to offer rigorous uncertainty quantification. Among these, conformal prediction has gained particular attention due to its ability to construct predictive regions with finite-sample validity under the mild assumption of independent and identically distributed (i.i.d.) data .

Despite its strong theoretical guarantees, standard conformal prediction primarily accounts for **aleatoric uncertainty**, which arises from inherent randomness in the data generation process. However, **epistemic uncertainty**, which captures uncertainty due to a lack of knowledge or data, is typically not well represented . This limitation can lead to misleadingly narrow predictive intervals in regions where the model has insufficient data, counterintuitive since higher uncertainty should correspond to wider intervals. We illustrate this issue with an example where standard conformal methods fail to expand intervals appropriately in regions with limited training data.

To address this shortcoming, we propose a novel approach that augments any conformal score with a measure of epistemic uncertainty. Our method ensures that predictive regions remain valid while properly reflecting both aleatoric and epistemic uncertainties, leading to more reliable uncertainty quantification in scenarios where data is sparse or non-representative.

### References

```bibtex

@article{wasserman2011frasian,
  title={Frasian inference},
  author={Wasserman, Larry},
  year={2011}
}

@misc{Manokhin2024,
  author = {Valery Manokhin},
  title = {Awesome Conformal Prediction},
  year = {2024},
  version = {1.0.0},
  doi = {10.5281/zenodo.6467204},
  url = {https://github.com/valeman/awesome-conformal-prediction}
}

@book{Izbicki2025,
  author    = {Rafael Izbicki},
  title     = {Machine Learning Beyond Point Predictions: Uncertainty Quantification},
  edition   = {1st},
  year      = {2025},
  pages     = {260},
  isbn      = {978-65-01-20272-3}
}


@misc{kaggle2016,
  author = "{Kaggle}",
  year = {2016},
  title = {House Sales in King County, USA},
  howpublished = {Online},
  url = {https://www.kaggle.com/harlfoxem/housesalesprediction/metadata},
  note = {Accessed: August, 2019}
}

@inproceedings{neshat2020optimisation,
  title={Optimisation of large wave farms using a multi-strategy evolutionary framework},
  author={Neshat, Mehdi and Alexander, Bradley and Sergiienko, Nataliia Y and Wagner, Markus},
  booktitle={Proceedings of the 2020 genetic and evolutionary computation conference},
  pages={1150--1158},
  year={2020}
}

@article{achilles2008tennessee,
  title={Tennessee’s student teacher achievement ratio (STAR) project},
  author={Achilles, Charles M and Bain, Helen Pate and Bellott, Fred and Boyd-Zaharias, Jayne and Finn, Jeremy and Folger, John and Johnston, John and Word, Elizabeth},
  journal={Harvard Dataverse},
  volume={1},
  pages={2008},
  year={2008}
}

@article{fanaee2014event,
  title={Event labeling combining ensemble detectors and background knowledge},
  author={Fanaee-T, Hadi and Gama, Joao},
  journal={Progress in Artificial Intelligence},
  volume={2},
  pages={113--127},
  year={2014},
  publisher={Springer}
}

@article{dua2017uci,
  title={UCI machine learning repository, 2017},
  author={Dua, Dheeru and Graff, Casey and others},
  journal={URL http://archive. ics. uci. edu/ml},
  volume={7},
  number={1},
  pages={62},
  year={2017}
}

@article{fong2021conformal,
  title={Conformal bayesian computation},
  author={Fong, Edwin and Holmes, Chris C},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={18268--18279},
  year={2021}
}



@article{he2023survey,
  title={A survey on uncertainty quantification methods for deep learning},
  author={He, Wenchong and Jiang, Zhe and Xiao, Tingsong and Xu, Zelin and Li, Yukun},
  journal={arXiv preprint arXiv:2302.13425},
  year={2023}
}

@book{gramacy2020surrogates,
  title={Surrogates: Gaussian process modeling, design, and optimization for the applied sciences},
  author={Gramacy, Robert B},
  year={2020},
  publisher={Chapman and Hall/CRC}
}

@article{quiroga2022bayesian,
  title={Bayesian additive regression trees for probabilistic programming},
  author={Quiroga, Miriana and Garay, Pablo G and Alonso, Juan M and Loyola, Juan Martin and Martin, Osvaldo A},
  journal={arXiv preprint arXiv:2206.03619},
  year={2022}
}

@article{kingma2014adam,
  title={Adam: A method for stochastic optimization},
  author={Kingma, Diederik P},
  journal={arXiv preprint arXiv:1412.6980},
  year={2014}
}

@article{kumar2017weight,
  title={On weight initialization in deep neural networks},
  author={Kumar, Siddharth Krishna},
  journal={arXiv preprint arXiv:1704.08863},
  year={2017}
}

@article{srivastava2014dropout,
  title={Dropout: a simple way to prevent neural networks from overfitting},
  author={Srivastava, Nitish and Hinton, Geoffrey and Krizhevsky, Alex and Sutskever, Ilya and Salakhutdinov, Ruslan},
  journal={The journal of machine learning research},
  volume={15},
  number={1},
  pages={1929--1958},
  year={2014},
  publisher={JMLR. org}
}

@inproceedings{ioffebatch2015,
author = {Ioffe, Sergey and Szegedy, Christian},
title = {Batch normalization: accelerating deep network training by reducing internal covariate shift},
year = {2015},
publisher = {JMLR.org},
booktitle = {Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37},
pages = {448–456},
numpages = {9},
location = {Lille, France},
series = {ICML'15}
}

@article{gardner2018gpytorch,
  title={Gpytorch: Blackbox matrix-matrix gaussian process inference with gpu acceleration},
  author={Gardner, Jacob and Pleiss, Geoff and Weinberger, Kilian Q and Bindel, David and Wilson, Andrew G},
  journal={Advances in neural information processing systems},
  volume={31},
  year={2018}
}

@article{feldman2021improving,
  title={Improving conditional coverage via orthogonal quantile regression},
  author={Feldman, Shai and Bates, Stephen and Romano, Yaniv},
  journal={Advances in neural information processing systems},
  volume={34},
  pages={2060--2071},
  year={2021}
}

@article{gneiting2007strictly,
  title={Strictly proper scoring rules, prediction, and estimation},
  author={Gneiting, Tilmann and Raftery, Adrian E},
  journal={Journal of the American statistical Association},
  volume={102},
  number={477},
  pages={359--378},
  year={2007},
  publisher={Taylor \& Francis}
}

@article{krizhevsky2009learning,
  title={Learning multiple layers of features from tiny images},
  author={Krizhevsky, Alex and Hinton, Geoffrey and others},
  year={2009},
  publisher={Toronto, ON, Canada}
}

@article{van2008visualizing,
  title={Visualizing data using t-SNE.},
  author={Van der Maaten, Laurens and Hinton, Geoffrey},
  journal={Journal of machine learning research},
  volume={9},
  number={11},
  year={2008}
}

@inproceedings{breunig2000lof,
  title={LOF: identifying density-based local outliers},
  author={Breunig, Markus M and Kriegel, Hans-Peter and Ng, Raymond T and Sander, J{\"o}rg},
  booktitle={Proceedings of the 2000 ACM SIGMOD international conference on Management of data},
  pages={93--104},
  year={2000}
}

@inproceedings{he2016deep,
  title={Deep residual learning for image recognition},
  author={He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian},
  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
  pages={770--778},
  year={2016}
}

@article{binois2018practical,
  title={Practical heteroscedastic Gaussian process modeling for large simulation experiments},
  author={Binois, Mickael and Gramacy, Robert B and Ludkovski, Mike},
  journal={Journal of Computational and Graphical Statistics},
  volume={27},
  number={4},
  pages={808--821},
  year={2018},
  publisher={Taylor \& Francis}
}

@inproceedings{salimbeni2018natural,
  title={Natural gradients in practice: Non-conjugate variational inference in Gaussian process models},
  author={Salimbeni, Hugh and Eleftheriadis, Stefanos and Hensman, James},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={689--697},
  year={2018},
  organization={PMLR}
}

@book{williams2006gaussian,
  title={Gaussian processes for machine learning},
  author={Williams, Christopher KI and Rasmussen, Carl Edward},
  volume={2},
  number={3},
  year={2006},
  publisher={MIT press Cambridge, MA}
}

@article{schulz2018tutorial,
  title={A tutorial on Gaussian process regression: Modelling, exploring, and exploiting functions},
  author={Schulz, Eric and Speekenbrink, Maarten and Krause, Andreas},
  journal={Journal of mathematical psychology},
  volume={85},
  pages={1--16},
  year={2018},
  publisher={Elsevier}
}

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

@article{pratola2020heteroscedastic,
  title={Heteroscedastic BART via multiplicative regression trees},
  author={Pratola, Matthew T and Chipman, Hugh A and George, Edward I and McCulloch, Robert E},
  journal={Journal of Computational and Graphical Statistics},
  volume={29},
  number={2},
  pages={405--417},
  year={2020},
  publisher={Taylor \& Francis}
}

@inproceedings{mcallister2017concrete,
  title={Concrete problems for autonomous vehicle safety: Advantages of Bayesian deep learning},
  author={McAllister, Rowan Thomas and Gal, Yarin and Kendall, Alex and Van Der Wilk, Mark and Shah, Amar and Cipolla, Roberto and Weller, Adrian},
  year={2017},
  organization={International Joint Conferences on Artificial Intelligence, Inc.}
}

@inproceedings{selvan2020uncertainty,
  title={Uncertainty quantification in medical image segmentation with normalizing flows},
  author={Selvan, Raghavendra and Faye, Frederik and Middleton, Jon and Pai, Akshay},
  booktitle={Machine Learning in Medical Imaging: 11th International Workshop, MLMI 2020, Held in Conjunction with MICCAI 2020, Lima, Peru, October 4, 2020, Proceedings 11},
  pages={80--90},
  year={2020},
  organization={Springer}
}


@article{vazquez2022conformal,
  title={Conformal prediction in clinical medical sciences},
  author={Vazquez, Janette and Facelli, Julio C},
  journal={Journal of Healthcare Informatics Research},
  volume={6},
  number={3},
  pages={241--252},
  year={2022},
  publisher={Springer}
}

@article{dorogush2018catboost,
  title={CatBoost: gradient boosting with categorical features support},
  author={Dorogush, Anna Veronika and Ershov, Vasily and Gulin, Andrey},
  journal={arXiv preprint arXiv:1810.11363},
  year={2018}
}

@article{sesia2020comparison,
  title={A comparison of some conformal quantile regression methods},
  author={Sesia, Matteo and Cand{\`e}s, Emmanuel J},
  journal={Stat},
  volume={9},
  number={1},
  pages={e261},
  year={2020},
  publisher={Wiley Online Library}
}


@article{wang2025aleatoric,
  title={From Aleatoric to Epistemic: Exploring Uncertainty Quantification Techniques in Artificial Intelligence},
  author={Wang, Tianyang and Wang, Yunze and Zhou, Jun and Peng, Benji and Song, Xinyuan and Zhang, Charles and Sun, Xintian and Niu, Qian and Liu, Junyu and Chen, Silin and others},
  journal={arXiv preprint arXiv:2501.03282},
  year={2025}
}


@article{tyralis2024review,
  title={A review of predictive uncertainty estimation with machine learning},
  author={Tyralis, Hristos and Papacharalampous, Georgia},
  journal={Artificial Intelligence Review},
  volume={57},
  number={4},
  pages={94},
  year={2024},
  publisher={Springer}
}




@article{huellermeier2021aleatoric,
  title={Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods},
  author={H{\"u}llermeier, Eyke and Waegeman, Willem},
  journal={Machine Learning},
  volume={110},
  number={3},
  pages={457--506},
  year={2021},
  publisher={Springer}
}
 
@article{chipman2010bart,
  title={{BART}: Bayesian additive regression trees},
  author={Chipman, Hugh A and George, Edward I and McCulloch, Robert E},
  journal={Annals of Applied Statistics},
  volume={6},
  number={1},
  pages={266--298},
  year={2012},
  publisher={Institute of Mathematical Statistics}
}



@article{fontana2023conformal,
  title={Conformal prediction: a unified review of theory and new challenges},
  author={Fontana, Matteo and Zeni, Gianluca and Vantini, Simone},
  journal={Bernoulli},
  volume={29},
  number={1},
  pages={1--23},
  year={2023},
  publisher={Bernoulli Society for Mathematical Statistics and Probability}
}

@inproceedings{bostrom2020mondrian,
  title={Mondrian conformal regressors},
  author={Bostr{\"o}m, Henrik and Johansson, Ulf},
  booktitle={Conformal and Probabilistic Prediction and Applications},
  pages={114--133},
  year={2020},
  organization={PMLR}
}


@article{guan2023localized,
  title={Localized conformal prediction: A generalized inference framework for conformal prediction},
  author={Guan, Leying},
  journal={Biometrika},
  volume={110},
  number={1},
  pages={33--50},
  year={2023},
  publisher={Oxford University Press}
}

@article{cabezas2024distribution,
  title={Distribution-Free Calibration of Statistical Confidence Sets},
  author={Cabezas, Luben and Soares, Guilherme P and Ramos, Thiago R and Stern, Rafael B and Izbicki, Rafael},
  journal={arXiv preprint arXiv:2411.19368},
  year={2024}
}


@article{cabezas2025regression,
  title={Regression trees for fast and adaptive prediction intervals},
  author={Cabezas, Luben MC and Otto, Mateus P and Izbicki, Rafael and Stern, Rafael B},
  journal={Information Sciences},
  volume={686},
  pages={121369},
  year={2025},
  publisher={Elsevier}
}


@article{lei2014distribution,
  title={Distribution-free prediction bands for non-parametric regression},
  author={Lei, Jing and Wasserman, Larry},
  journal={Journal of the Royal Statistical Society Series B: Statistical Methodology},
  volume={76},
  number={1},
  pages={71--96},
  year={2014},
  publisher={Oxford University Press}
}



@inproceedings{pion2024gaussian,
  title={Gaussian process interpolation with conformal prediction: methods and comparative analysis},
  author={Pion, Aur{\'e}lien and Vazquez, Emmanuel},
  booktitle={LOD 2024, 10th International Conference on Machine Learning, Optimization, and Data Science},
  year={2024}
}

@article{dheur2025multi,
  title={Multi-Output Conformal Regression: A Unified Comparative Study with New Conformity Scores},
  author={Dheur, Victor and Fontana, Matteo and Estievenart, Yorick and Desobry, Naomi and Taieb, Souhaib Ben},
  journal={arXiv preprint arXiv:2501.10533},
  year={2025}
}

@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}
}

@inproceedings{angelopoulos2021uncertainty,
  title={Uncertainty Sets for Image Classifiers using Conformal Prediction},
  author={Anastasios Nikolas Angelopoulos and Stephen Bates and Jitendra Malik and Michael I. Jordan},
  booktitle={International Conference on Learning Representations},
  year={2021},
  url={https://openreview.net/forum?id=eNdiU_DbM9}
}


@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},
	year = {2016},
	series = {Proceedings of Machine Learning Research},
	pages = {1050-1059},
	publisher = {PMLR}
}

@article{romano2020classification,
  title={Classification with valid and adaptive coverage},
  author={Romano, Yaniv and Sesia, Matteo and Candes, Emmanuel},
  journal={Advances in Neural Information Processing Systems},
  volume={33},
  pages={3581--3591},
  year={2020}
}



@article{valle2023quantifying,
  title={Quantifying uncertainty in land-use land-cover classification using conformal statistics},
  author={Valle, Denis and Izbicki, Rafael and Leite, Rodrigo Vieira},
  journal={Remote Sensing of Environment},
  volume={295},
  pages={113682},
  year={2023},
  publisher={Elsevier}
}


@inproceedings{papadopoulos2002inductive,
  title={Inductive confidence machines for regression},
  author={Papadopoulos, Harris and Proedrou, Kostas and Vovk, Volodya and Gammerman, Alex},
  booktitle={Machine learning: ECML 2002: 13th European conference on machine learning Helsinki, Finland, August 19--23, 2002 proceedings 13},
  pages={345--356},
  year={2002},
  organization={Springer}
}


@article{foygel2021limits,
  title={The limits of distribution-free conditional predictive inference},
  author={Foygel Barber, Rina and Candes, Emmanuel J and Ramdas, Aaditya and Tibshirani, Ryan J},
  journal={Information and Inference: A Journal of the IMA},
  volume={10},
  number={2},
  pages={455--482},
  year={2021},
  publisher={Oxford University Press}
}



@article{jaber2024conformal,
  title={Conformal Approach To Gaussian Process Surrogate Evaluation With Coverage Guarantees},
  author={Jaber, Edgar and Blot, Vincent and Brunel, Nicolas and Chabridon, Vincent and Remy, Emmanuel and Iooss, Bertrand and Lucor, Didier and Mougeot, Mathilde and Leite, Alessandro},
  journal={arXiv preprint arXiv:2401.07733},
  year={2024}
}

@inproceedings{bethell2024robust,
  title={Robust Uncertainty Quantification Using Conformalised Monte Carlo Prediction},
  author={Bethell, Daniel and Gerasimou, Simos and Calinescu, Radu},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={38},
  number={19},
  pages={20939--20948},
  year={2024}
}

@article{cocheteux2025uncertainty,
  title={Uncertainty-Aware Online Extrinsic Calibration: A Conformal Prediction Approach},
  author={Cocheteux, Mathieu and Moreau, Julien and Davoine, Franck},
  journal={arXiv preprint arXiv:2501.06878},
  year={2025}
}

@article{hullermeier2021aleatoric,
  title={Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods},
  author={H{\"u}llermeier, Eyke and Waegeman, Willem},
  journal={Machine Learning},
  volume={110},
  pages={457--506},
  year={2021},
  publisher={Springer}
}

@book{schervish2012theory,
  title={Theory of statistics},
  author={Schervish, Mark J},
  year={2012},
  publisher={Springer Science \& Business Media}
}

@article{freeman2017unified,
  title={A unified framework for constructing, tuning and assessing photometric redshift density estimates in a selection bias setting},
  author={Freeman, Peter E and Izbicki, Rafael and Lee, Ann B},
  journal={Monthly Notices of the Royal Astronomical Society},
  volume={468},
  number={4},
  pages={4556--4565},
  year={2017},
  publisher={Oxford University Press}
}

@article{izbicki2017converting,
  title={Converting high-dimensional regression to high-dimensional conditional density estimation},
  author={Izbicki, Rafael and Lee, Ann B},
  journal={Electronic Journal of Statistics},
  volume={11},
  pages={2800--2831},
  year={2017}
}

@article{barber2021predictive,
  title={Predictive inference with the jackknife+},
  author={Barber, Rina Foygel and Candes, Emmanuel J and Ramdas, Aaditya and Tibshirani, Ryan J},
  journal={The Annals of Statistics},
  volume={49},
  number={1},
  pages={486--507},
  year={2021},
  publisher={JSTOR}
}

@article{pedregosa2011scikit,
  title={Scikit-learn: Machine learning in Python},
  author={Pedregosa, Fabian and Varoquaux, Ga{\"e}l and Gramfort, Alexandre and Michel, Vincent and Thirion, Bertrand and Grisel, Olivier and Blondel, Mathieu and Prettenhofer, Peter and Weiss, Ron and Dubourg, Vincent and others},
  journal={the Journal of machine Learning research},
  volume={12},
  pages={2825--2830},
  year={2011},
  publisher={JMLR. org}
}

@incollection{frazier2018bayesian,
  title={Bayesian optimization},
  author={Frazier, Peter I},
  booktitle={Recent advances in optimization and modeling of contemporary problems},
  pages={255--278},
  year={2018},
  publisher={Informs}
}

@ARTICLE{Schmidt2020Photo-z,
       author = {{Schmidt}, S.~J. and {Malz}, A.~I. and {Soo}, J.~Y.~H. and {Almosallam}, I.~A. and {Brescia}, M. and {Cavuoti}, S. and {Cohen-Tanugi}, J. and {Connolly}, A.~J. and {DeRose}, J. and {Freeman}, P.~E. and {Graham}, M.~L. and {Iyer}, K.~G. and {Jarvis}, M.~J. and {Kalmbach}, J.~B. and {Kovacs}, E. and {Lee}, A.~B. and {Longo}, G. and {Morrison}, C.~B. and {Newman}, J.~A. and {Nourbakhsh}, E. and {Nuss}, E. and {Pospisil}, T. and {Tranin}, H. and {Wechsler}, R.~H. and {Zhou}, R. and {Izbicki}, R. and {LSST Dark Energy Science Collaboration}},
        title = "{Evaluation of probabilistic photometric redshift estimation approaches for The Rubin Observatory Legacy Survey of Space and Time (LSST)}",
      journal = {Monthly Notices of the Royal Astronomical Society},
     keywords = {methods: statistical, galaxies: distances and redshifts, galaxies: statistics, Astrophysics - Cosmology and Nongalactic Astrophysics, Astrophysics - Instrumentation and Methods for Astrophysics},
         year = 2020,
        month = dec,
       volume = {499},
       number = {2},
        pages = {1587-1606},
archivePrefix = {arXiv},
       eprint = {2001.03621},
 primaryClass = {astro-ph.CO},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2020MNRAS.499.1587S},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}


@article{chernozhukov2021distributional,
  title={Distributional conformal prediction},
  author={Chernozhukov, Victor and W{\"u}thrich, Kaspar and Zhu, Yinchu},
  journal={Proceedings of the National Academy of Sciences},
  volume={118},
  number={48},
  pages={e2107794118},
  year={2021},
  publisher={National Acad Sciences}
}


@inproceedings{izbicki2020flexible,
  title={Flexible distribution-free conditional predictive bands using density estimators},
  author={Izbicki, Rafael and Shimizu, Gilson and Stern, Rafael},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={3068--3077},
  year={2020},
  organization={PMLR}
}

@article{plassier2024conditionally,
  title={Conditionally valid Probabilistic Conformal Prediction},
  author={Plassier, Vincent and Fishkov, Alexander and Panov, Maxim and Moulines, Eric},
  journal={stat},
  volume={1050},
  pages={1},
  year={2024}
}


@inproceedings{rossellini2024integrating,
  title={Integrating uncertainty awareness into conformalized quantile regression},
  author={Rossellini, Raphael and Barber, Rina Foygel and Willett, Rebecca},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={1540--1548},
  year={2024},
  organization={PMLR}
}



@article{izbicki2022cd,
  title={Cd-split and hpd-split: Efficient conformal regions in high dimensions},
  author={Izbicki, Rafael and Shimizu, Gilson and Stern, Rafael B},
  journal={Journal of Machine Learning Research},
  volume={23},
  number={87},
  pages={1--32},
  year={2022}
}



@inproceedings{frohlich2025personalizedus,
  title={PersonalizedUS: Interpretable Breast Cancer Risk Assessment with Local Coverage Uncertainty Quantification},
  author={Fr{\"o}hlich, Alek and Ramos, Thiago and Dos Santos, Gustavo Motta Cabello and Buzatto, Isabela Panzeri Carlotti and Izbicki, Rafael and Tiezzi, Daniel Guimaraes},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={39},
  number={27},
  pages={27998--28006},
  year={2025}
}


@article{valle2024nonparametric,
  title = {Nonparametric quantification of uncertainty in multistep upscaling approaches: a case study on estimating forest biomass in the Brazilian Amazon},
  author = {Valle, D. and Haneda, L. and Izbicki, R. and Kamimura, R. and Azevedo, B. and Gomes, S. and Sanchez, A. and Almeida, D.},
  journal = {Science of Remote Sensing},
  year = {2024},
  note = {To appear}
}


@book{vovk2005algorithmic,
  title={Algorithmic learning in a random world},
  author={Vovk, Vladimir and Gammerman, Alexander and Shafer, Glenn},
  volume={29},
  year={2005},
  publisher={Springer}
}

@inproceedings{romano2019,
	author = {Romano, Yaniv and Patterson, Evan and Cand\`{e}s, Emmanuel},
	booktitle = {Advances in Neural Information Processing Systems},
	editor = {},
	pages = {3543-3553},
	publisher = {Curran Associates, Inc.},
	title = {{Conformalized quantile regression}},
	volume = {32},
	year = {2019}
}

@article{Lei2018,
	title={Distribution-free predictive inference for regression},
	author={Lei, Jing and G'Sell, Max and Rinaldo, Alessandro and Tibshirani, Ryan J and Wasserman, Larry},
	journal={Journal of the American Statistical Association},
	volume={113},
	number={523},
	pages={1094--1111},
	year={2018},
	publisher={Taylor \& Francis}
}

@article{shafer2008tutorial,
  title={A Tutorial on Conformal Prediction.},
  author={Shafer, Glenn and Vovk, Vladimir},
  journal={Journal of Machine Learning Research},
  volume={9},
  number={3},
  year={2008}
}


@article{Valle2024,
  author    = {D. Valle and R. Leite and Rafael Izbicki and C. Silva and L. Haneda},
  title     = {Local uncertainty maps for land-use/land-cover classification without remote sensing and modeling work using a class-conditional conformal approach},
  journal   = {International Journal of Applied Earth Observation and Geoinformation},
  year      = {2024}
}



@article{horta2015potential,
  title={Potential of integrated field spectroscopy and spatial analysis for enhanced assessment of soil contamination: A prospective review},
  author={Horta, Ana and Malone, B and Stockmann, U and Minasny, B and Bishop, TFA and McBratney, AB and Pallasser, R and Pozza, L},
  journal={Geoderma},
  volume={241},
  pages={180--209},
  year={2015},
  publisher={Elsevier}
}



 @article{mian2024literature,
  title={A literature review of fault diagnosis based on ensemble learning},
  author={Mian, Zhibao and Deng, Xiaofei and Dong, Xiaohui and Tian, Yuzhu and Cao, Tianya and Chen, Kairan and Al Jaber, Tareq},
  journal={Engineering Applications of Artificial Intelligence},
  volume={127},
  pages={107357},
  year={2024},
  publisher={Elsevier}
}


@inproceedings{csillag2023amnioml,
  title={AmnioML: amniotic fluid segmentation and volume prediction with uncertainty quantification},
  author={Csillag, Daniel and Paes, Lucas Monteiro and Ramos, Thiago and Romano, Jo{\~a}o Vitor and Schuller, Rodrigo and Seixas, Roberto B and Oliveira, Roberto I and Orenstein, Paulo},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={37},
  number={13},
  pages={15494--15502},
  year={2023}
}


@article{angelopoulos2021gentle,
  title={A gentle introduction to conformal prediction and distribution-free uncertainty quantification},
  author={Angelopoulos, Anastasios N and Bates, Stephen},
  journal={arXiv preprint arXiv:2107.07511},
  year={2021}
}


%%% DATASETS
@misc{concrete_compressive_strength_165,
  author       = {Yeh, I-Cheng},
  title        = {{Concrete Compressive Strength}},
  year         = {1998},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C5PK67}
}

@misc{wine_quality_186,
  author       = {Cortez, Paulo and Cerdeira, A. and Almeida, F. and Matos, T. and Reis, J.},
  title        = {{Wine Quality}},
  year         = {2009},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C56S3T}
}

@misc{electrical_grid_stability_simulated_data__471,
  author       = {Arzamasov, Vadim},
  title        = {{Electrical Grid Stability Simulated Data }},
  year         = {2018},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C5PG66}
}

@misc{combined_cycle_power_plant_294,
  author       = {Tfekci, Pnar and Kaya, Heysem},
  title        = {{Combined Cycle Power Plant}},
  year         = {2014},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C5002N}
}

@misc{kaggle_bike_sharing_demand,
  author       = {Raj Mehra},
  title        = {Bike Sharing Demand - RMSLE: 0.3194},
  year         = {2023},
  howpublished = {\url{https://www.kaggle.com/code/rajmehra03/bike-sharing-demand-rmsle-0-3194/input?select=train.csv}},
  note         = {Accessed: [Insert Date Here]},
  publisher    = {Kaggle},
}
@misc{github_clover_meps,
  author       = {Monoxido45},
  title        = {MEPS Dataset in Clover Repository},
  year         = {2023},
  howpublished = {\url{https://github.com/Monoxido45/clover/tree/main/data/raw/meps}},
  note         = {Accessed: [Insert Date Here]},
  publisher    = {GitHub},
}

@misc{superconductivty_data_464,
  author       = {Hamidieh, Kam},
  title        = {{Superconductivty Data}},
  year         = {2018},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C53P47}
}

@misc{online_news_popularity_332,
  author       = {Kelwin Fernandes and  Pedro Vinagre and  Paulo Cortez and  Pedro Sernadela},
  title        = {{Online News Popularity}},
  year         = {2015},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C5NS3V}
}

@misc{physicochemical_properties_of_protein_tertiary_structure_265,
  author       = {Rana, Prashant},
  title        = {{Physicochemical Properties of Protein Tertiary Structure}},
  year         = {2013},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C5QW3H}
}

@article{good1992bayes,
  title={The Bayes/non-Bayes compromise: A brief review},
  author={Good, Irving John},
  journal={Journal of the American Statistical Association},
  volume={87},
  number={419},
  pages={597--606},
  year={1992},
  publisher={Taylor \& Francis}
}


@book{bernardo2009bayesian,
  title={Bayesian Theory},
  author={Bernardo, J.M. and Smith, A.F.M.},
  isbn={9780470317716},
  series={Wiley Series in Probability and Statistics},
  url={https://books.google.com.br/books?id=11nSgIcd7xQC},
  year={2009},
  publisher={Wiley}
}

@InProceedings{Gal2016,
  title = 	 {Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning},
  author = 	 {Gal, Yarin and Ghahramani, Zoubin},
  booktitle = 	 {Proceedings of The 33rd International Conference on Machine Learning},
  pages = 	 {1050--1059},
  year = 	 {2016},
  editor = 	 {Balcan, Maria Florina and Weinberger, Kilian Q.},
  volume = 	 {48},
  series = 	 {Proceedings of Machine Learning Research},
  address = 	 {New York, New York, USA},
  month = 	 {20--22 Jun},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v48/gal16.pdf},
  url = 	 {https://proceedings.mlr.press/v48/gal16.html}
}

@InProceedings{pmlr-v31-damianou13a,
  title = 	 {Deep {G}aussian Processes},
  author = 	 {Damianou, Andreas and Lawrence, Neil D.},
  booktitle = 	 {Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics},
  pages = 	 {207--215},
  year = 	 {2013},
  editor = 	 {Carvalho, Carlos M. and Ravikumar, Pradeep},
  volume = 	 {31},
  series = 	 {Proceedings of Machine Learning Research},
  address = 	 {Scottsdale, Arizona, USA},
  month = 	 {29 Apr--01 May},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v31/damianou13a.pdf},
  url = 	 {https://proceedings.mlr.press/v31/damianou13a.html}
}

@article{hoff2023bayes,
  title={Bayes-optimal prediction with frequentist coverage control},
  author={Hoff, Peter},
  journal={Bernoulli},
  volume={29},
  number={2},
  pages={901--928},
  year={2023},
  publisher={Bernoulli Society for Mathematical Statistics and Probability}
}


@article{dalmasso2024likelihood,
  title={Likelihood-free frequentist inference: Bridging classical statistics and machine learning for reliable simulator-based inference},
  author={Dalmasso, Niccol{\`o} and Masserano, Luca and Zhao, David and Izbicki, Rafael and Lee, Ann B},
  journal={Electronic Journal of Statistics},
  volume={18},
  number={2},
  pages={5045--5090},
  year={2024},
  publisher={The Institute of Mathematical Statistics and the Bernoulli Society}
}

@inproceedings{masserano2023simulator,
  title={Simulator-Based Inference with WALDO: Confidence Regions by Leveraging Prediction Algorithms and Posterior Estimators for Inverse Problems},
  author={Masserano, Luca and Dorigo, Tommaso and Izbicki, Rafael and Kuusela, Mikael and Lee, Ann},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={2960--2974},
  year={2023},
  organization={PMLR}
}





``` 

