@article{10.1111/ectj.12097,
  author   = {Chernozhukov, Victor and Chetverikov, Denis and Demirer, Mert and Duflo, Esther and Hansen, Christian and Newey, Whitney and Robins, James},
  title    = {{Double/debiased machine learning for treatment and structural parameters}},
  journal  = {The Econometrics Journal},
  volume   = {21},
  number   = {1},
  pages    = {C1-C68},
  year     = {2018},
  month    = {01},
  abstract = {{We revisit the classic semi‐parametric problem of inference on a low‐dimensional parameter θ0 in the presence of high‐dimensional nuisance parameters η0. We depart from the classical setting by allowing for η0 to be so high‐dimensional that the traditional assumptions (e.g. Donsker properties) that limit complexity of the parameter space for this object break down. To estimate η0, we consider the use of statistical or machine learning (ML) methods, which are particularly well suited to estimation in modern, very high‐dimensional cases. ML methods perform well by employing regularization to reduce variance and trading off regularization bias with overfitting in practice. However, both regularization bias and overfitting in estimating η0 cause a heavy bias in estimators of θ0 that are obtained by naively plugging ML estimators of η0 into estimating equations for θ0. This bias results in the naive estimator failing to be N−1/2 consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest θ0 can be removed by using two simple, yet critical, ingredients: (1) using Neyman‐orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ0; (2) making use of cross‐fitting, which provides an efficient form of data‐splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in an N−1/2‐neighbourhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid confidence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements, which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters, such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of the following: DML applied to learn the main regression parameter in a partially linear regression model; DML applied to learn the coefficient on an endogenous variable in a partially linear instrumental variables model; DML applied to learn the average treatment effect and the average treatment effect on the treated under unconfoundedness; DML applied to learn the local average treatment effect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.}}
}
@inproceedings{benkeser2016highly,
  title        = {The highly adaptive lasso estimator},
  author       = {Benkeser, David and {v}an {d}er Laan, Mark},
  booktitle    = {2016 IEEE international conference on data science and advanced analytics (DSAA)},
  pages        = {689--696},
  year         = {2016},
  organization = {IEEE}
}
@book{bickel1993efficient,
  title     = {Efficient and adaptive estimation for semiparametric models},
  author    = {Bickel, Peter J and Klaassen, Chris AJ and Bickel, Peter J and Ritov, Ya’acov and Klaassen, J and Wellner, Jon A and Ritov, YA'Acov},
  volume    = {4},
  year      = {1993},
  publisher = {Springer}
}
@article{bickler2021machine,
  title     = {Machine learning arrives in archaeology},
  author    = {Bickler, Simon H},
  journal   = {Advances in Archaeological Practice},
  volume    = {9},
  number    = {2},
  pages     = {186--191},
  year      = {2021},
  publisher = {Cambridge University Press}
}
@misc{bike_sharing_275,
  author       = {Fanaee-T, Hadi},
  title        = {{Bike Sharing}},
  year         = {2013},
  howpublished = {UCI Machine Learning Repository},
  note         = {{DOI}: https://doi.org/10.24432/C5W894}
}
@article{booth1998monte,
  title     = {Monte Carlo approximation of bootstrap variances},
  author    = {Booth, James G and Sarkar, Somnath},
  journal   = {The American Statistician},
  volume    = {52},
  number    = {4},
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}
@article{breiman2001random,
  title     = {Random forests},
  author    = {Breiman, Leo},
  journal   = {Machine learning},
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  publisher = {Springer}
}

@article{cai2020one,
  title     = {One-step targeted maximum likelihood estimation for time-to-event outcomes},
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  volume    = {76},
  number    = {3},
  pages     = {722--733},
  year      = {2020},
  publisher = {Oxford University Press}
}
@article{chamma2024statistically,
  title   = {Statistically valid variable importance assessment through conditional permutations},
  author  = {Chamma, Ahmad and Engemann, Denis A and Thirion, Bertrand},
  journal = {Advances in Neural Information Processing Systems},
  volume  = {36},
  year    = {2024}
}
@article{cheng1984strong,
  title     = {Strong consistency of nearest neighbor regression function estimators},
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@inproceedings{cho2023kernel,
  title     = {Kernel Debiased Plug-in Estimation: Simultaneous, Automated Debiasing without Influence Functions for Many Target Parameters},
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  booktitle = {Forty-first International Conference on Machine Learning},
  year      = {2023}
}
@article{diciccio1996bootstrap,
  title     = {Bootstrap confidence intervals},
  author    = {DiCiccio, Thomas J and Efron, Bradley},
  journal   = {Statistical science},
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  number    = {3},
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  title   = {The rashomon importance distribution: Getting rid of unstable, single model-based variable importance},
  author  = {Donnelly, Jon and Katta, Srikar and Rudin, Cynthia and Browne, Edward},
  journal = {Advances in Neural Information Processing Systems},
  volume  = {36},
  pages   = {6267--6279},
  year    = {2023}
}
@article{efron1985bootstrap,
  title     = {Bootstrap confidence intervals for a class of parametric problems},
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  journal   = {Biometrika},
  volume    = {72},
  number    = {1},
  pages     = {45--58},
  year      = {1985},
  publisher = {Oxford University Press}
}


@article{farrell2018deep,
  title   = {Deep neural networks for estimation and inference: application to causal effects and other semiparametric estimands},
  author  = {Farrell, Max H and Liang, Tengyuan and Misra, Sanjog},
  journal = {arXiv preprint arXiv:1809.09953},
  volume  = {20},
  year    = {2018}
}

@article{fauvel2025sobol,
  title   = {Sobol-CPI: a Doubly Robust Conditional Permutation Importance Statistic},
  author  = {Fauvel, Kevin and Morvan, Marine and Cadre, Baptiste and Subtil, François},
  journal = {arXiv preprint arXiv:2501.17520},
  year    = {2025}
}
@article{fisher2019all,
  title   = {All models are wrong, but many are useful: Learning a variable's importance by studying an entire class of prediction models simultaneously},
  author  = {Fisher, Aaron and Rudin, Cynthia and Dominici, Francesca},
  journal = {Journal of Machine Learning Research},
  volume  = {20},
  number  = {177},
  pages   = {1--81},
  year    = {2019}
}

@article{freiesleben2024scientific,
  title     = {Scientific Inference with Interpretable Machine Learning: Analyzing Models to Learn About Real-World Phenomena},
  author    = {Freiesleben, Timo and K{\"o}nig, Gunnar and Molnar, Christoph and Tejero-Cantero, {\'A}lvaro},
  journal   = {Minds and Machines},
  volume    = {34},
  number    = {3},
  pages     = {32},
  year      = {2024},
  publisher = {Springer}
}
@article{gan2022model,
  title   = {Model-agnostic confidence intervals for feature importance: A fast and powerful approach using minipatch ensembles},
  author  = {Gan, Luqin and Zheng, Lili and Allen, Genevera I},
  journal = {arXiv preprint arXiv:2206.02088},
  year    = {2022}
}
@article{gromping2007estimators,
  title     = {Estimators of relative importance in linear regression based on variance decomposition},
  author    = {Gr{\"o}mping, Ulrike},
  journal   = {The American Statistician},
  volume    = {61},
  number    = {2},
  pages     = {139--147},
  year      = {2007},
  publisher = {Taylor \& Francis}
}
@article{gruber2010targeted,
  title     = {A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome},
  author    = {Gruber, Susan and van der Laan, Mark J},
  journal   = {The International Journal of Biostatistics},
  volume    = {6},
  number    = {1},
  year      = {2010},
  publisher = {De Gruyter}
}
@article{hines2022demystifying,
  title     = {Demystifying statistical learning based on efficient influence functions},
  author    = {Hines, Oliver and Dukes, Oliver and Diaz-Ordaz, Karla and Vansteelandt, Stijn},
  journal   = {The American Statistician},
  volume    = {76},
  number    = {3},
  pages     = {292--304},
  year      = {2022},
  publisher = {Taylor \& Francis}
}

@article{hooker2017machine,
  title   = {Machine learning and the future of realism},
  author  = {Hooker, Giles and Hooker, Cliff},
  journal = {arXiv preprint arXiv:1704.04688},
  year    = {2017}
}
@article{hooker2021unrestricted,
  title     = {Unrestricted permutation forces extrapolation: variable importance requires at least one more model, or there is no free variable importance},
  author    = {Hooker, Giles and Mentch, Lucas and Zhou, Siyu},
  journal   = {Statistics and Computing},
  volume    = {31},
  pages     = {1--16},
  year      = {2021},
  publisher = {Springer}
}
@article{kennedy2022semiparametric,
  title   = {Semiparametric doubly robust targeted double machine learning: a review},
  author  = {Kennedy, Edward H},
  journal = {arXiv preprint arXiv:2203.06469},
  year    = {2022}
}
@article{lei2018distribution,
  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{lu2021unified,
  title   = {A unified framework for random forest prediction error estimation},
  author  = {Lu, Benjamin and Hardin, Johanna},
  journal = {Journal of Machine Learning Research},
  volume  = {22},
  number  = {8},
  pages   = {1--41},
  year    = {2021}
}
@article{lu2023using,
  title     = {Using machine learning to predict urban canopy flows for land surface modeling},
  author    = {Lu, Yanle and Zhou, Xu-Hui and Xiao, Heng and Li, Qi},
  journal   = {Geophysical Research Letters},
  volume    = {50},
  number    = {1},
  pages     = {e2022GL102313},
  year      = {2023},
  publisher = {Wiley Online Library}
}
@article{luedtke2016optimal,
  title     = {Optimal targeted learning: confidence intervals for a median parameter},
  author    = {Luedtke, Alexander R and {v}an {d}er Laan, Mark J},
  journal   = {Statistical Methods in Medical Research},
  volume    = {25},
  number    = {3},
  pages     = {897--917},
  year      = {2016},
  publisher = {SAGE Publications Sage UK: London, England}
}
@inproceedings{lundberg2017unified,
  title     = {A Unified Approach to Interpreting Model Predictions},
  author    = {Lundberg, Scott M and Lee, Su-In},
  booktitle = {Proceedings of the 31st International Conference on Neural Information Processing Systems (NeurIPS)},
  pages     = {4768--4777},
  year      = {2017}
}
@article{luo2016high,
  title   = {High-Dimensional $ L\_2 $ Boosting: Rate of Convergence},
  author  = {Luo, Ye and Spindler, Martin and K{\"u}ck, Jannis},
  journal = {arXiv preprint arXiv:1602.08927},
  year    = {2016}
}

@article{mentch2016quantifying,
  title   = {Quantifying uncertainty in random forests via confidence intervals and hypothesis tests},
  author  = {Mentch, Lucas and Hooker, Giles},
  journal = {Journal of Machine Learning Research},
  volume  = {17},
  number  = {26},
  pages   = {1--41},
  year    = {2016}
}
@article{molina2019machine,
  title     = {Machine learning for sociology},
  author    = {Molina, Mario and Garip, Filiz},
  journal   = {Annual Review of Sociology},
  volume    = {45},
  number    = {1},
  pages     = {27--45},
  year      = {2019},
  publisher = {Annual Reviews}
}
@book{molnar2020interpretable,
  title     = {Interpretable machine learning},
  author    = {Molnar, Christoph},
  year      = {2020},
  publisher = {Lulu. com}
}

@inproceedings{molnar2023relating,
  title        = {Relating the partial dependence plot and permutation feature importance to the data generating process},
  author       = {Molnar, Christoph and Freiesleben, Timo and K{\"o}nig, Gunnar and Herbinger, Julia and Reisinger, Tim and Casalicchio, Giuseppe and Wright, Marvin N and Bischl, Bernd},
  booktitle    = {World Conference on Explainable Artificial Intelligence},
  pages        = {456--479},
  year         = {2023},
  organization = {Springer}
}
@article{newey2018cross,
  title   = {Cross-fitting and fast remainder rates for semiparametric estimation},
  author  = {Newey, Whitney K and Robins, James R},
  journal = {arXiv preprint arXiv:1801.09138},
  year    = {2018}
}
@article{ning2017general,
  title   = {A general theory of hypothesis tests and confidence regions for sparse high dimensional models},
  author  = {Ning, Yang and Liu, Han},
  year    = {2017},
  journal = {The Annals of Statistics}
}
@inproceedings{ribeiro2016should,
  title     = {"Why Should I Trust You?": Explaining the Predictions of Any Classifier},
  author    = {Ribeiro, Marco Tulio and Singh, Sameer and Guestrin, Carlos},
  booktitle = {Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD)},
  pages     = {1135--1144},
  year      = {2016}
}
@article{scott2017unified,
  title     = {A unified approach to interpreting model predictions},
  author    = {Scott, M and Su-In, Lee and others},
  journal   = {Advances in neural information processing systems},
  volume    = {30},
  pages     = {4765--4774},
  year      = {2017},
  publisher = {Curran Associates, Inc}
}
@article{smith2023application,
  title     = {Application of targeted maximum likelihood estimation in public health and epidemiological studies: a systematic review},
  author    = {Smith, Matthew J and Phillips, Rachael V and Luque-Fernandez, Miguel Angel and Maringe, Camille},
  journal   = {Annals of epidemiology},
  volume    = {86},
  pages     = {34--48},
  year      = {2023},
  publisher = {Elsevier}
}
@article{smith2023application,
  title     = {Application of targeted maximum likelihood estimation in public health and epidemiological studies: a systematic review},
  author    = {Smith, Matthew J and Phillips, Rachael V and Luque-Fernandez, Miguel Angel and Maringe, Camille},
  journal   = {Annals of epidemiology},
  volume    = {86},
  pages     = {34--48},
  year      = {2023},
  publisher = {Elsevier}
}
@article{strobl2008conditional,
  title     = {Conditional variable importance for random forests},
  author    = {Strobl, Carolin and Boulesteix, Anne-Laure and Kneib, Thomas and Augustin, Thomas and Zeileis, Achim},
  journal   = {BMC bioinformatics},
  volume    = {9},
  pages     = {1--11},
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  publisher = {Springer}
}
@book{tsiatis2006semiparametric,
  title     = {Semiparametric theory and missing data},
  author    = {Tsiatis, Anastasios A},
  volume    = {4},
  year      = {2006},
  publisher = {Springer}
}

@book{van2000asymptotic,
  title     = {Asymptotic statistics},
  author    = {{v}an {d}er Vaart, Aad W},
  volume    = {3},
  year      = {2000},
  publisher = {Cambridge university press}
} 

@article{van2006statistical,
  title     = {Statistical inference for variable importance},
  author    = {{v}an {d}er Laan, Mark J},
  journal   = {The International Journal of Biostatistics},
  volume    = {2},
  number    = {1},
  year      = {2006},
  publisher = {De Gruyter}
}

@article{van2006targeted,
  title     = {Targeted maximum likelihood learning},
  author    = {{v}an {d}er Laan, Mark J and Rubin, Daniel},
  journal   = {The international journal of biostatistics},
  volume    = {2},
  number    = {1},
  year      = {2006},
  publisher = {De Gruyter}
}
@article{van2011cross,
  title     = {Cross-validated targeted minimum-loss-based estimation},
  author    = {{v}an {d}er Laan, Mark J and Rose, Sherri and Zheng, Wenjing},
  journal   = {Targeted learning: causal inference for observational and experimental data},
  pages     = {459--474},
  year      = {2011},
  publisher = {Springer}
}
@book{van2011targeted,
  title     = {Targeted learning: causal inference for observational and experimental data},
  author    = {{v}an {d}er Laan, Mark J and Rose, Sherri and others},
  volume    = {4},
  year      = {2011},
  publisher = {Springer}
}
@article{van2016one,
  title     = {One-step targeted minimum loss-based estimation based on universal least favorable one-dimensional submodels},
  author    = {{v}an {d}er Laan, Mark and Gruber, Susan},
  journal   = {The international journal of biostatistics},
  volume    = {12},
  number    = {1},
  pages     = {351--378},
  year      = {2016},
  publisher = {De Gruyter}
}

@article{van2024automatic,
  title   = {Automatic doubly robust inference for linear functionals via calibrated debiased machine learning},
  author  = {{v}an {d}er Laan, Lars and Luedtke, Alex and Carone, Marco},
  journal = {arXiv preprint arXiv:2411.02771},
  year    = {2024}
}

@article{verdinelli2024decorrelated,
  title   = {Decorrelated variable importance},
  author  = {Verdinelli, Isabella and Wasserman, Larry},
  journal = {Journal of Machine Learning Research},
  volume  = {25},
  number  = {7},
  pages   = {1--27},
  year    = {2024}
}
@article{verdinelli2024feature,
  title     = {Feature importance: A closer look at shapley values and loco},
  author    = {Verdinelli, Isabella and Wasserman, Larry},
  journal   = {Statistical Science},
  volume    = {39},
  number    = {4},
  pages     = {623--636},
  year      = {2024},
  publisher = {Institute of Mathematical Statistics}
}

@article{wei2023efficient,
  title     = {Efficient targeted learning of heterogeneous treatment effects for multiple subgroups},
  author    = {Wei, Waverly and Petersen, Maya and {v}an {d}er Laan, Mark J and Zheng, Zeyu and Wu, Chong and Wang, Jingshen},
  journal   = {Biometrics},
  volume    = {79},
  number    = {3},
  pages     = {1934--1946},
  year      = {2023},
  publisher = {Wiley Online Library}
}

@article{wellner2006semiparametric,
  title   = {Semiparametric models: a review of progress since BKRW (1993)},
  author  = {Wellner, Jon A and Klaassen, Chris AJ and Ritov, Ya’acov and others},
  journal = {Frontiers in statistics. Imperial College Press, London},
  pages   = {25--44},
  year    = {2006}
}

@inproceedings{williamson2020efficient,
  title        = {Efficient nonparametric statistical inference on population feature importance using Shapley values},
  author       = {Williamson, Brian and Feng, Jean},
  booktitle    = {International conference on machine learning},
  pages        = {10282--10291},
  year         = {2020},
  organization = {PMLR}
}
@article{williamson2021nonparametric,
  title     = {Nonparametric variable importance assessment using machine learning techniques},
  author    = {Williamson, Brian D and Gilbert, Peter B and Carone, Marco and Simon, Noah},
  journal   = {Biometrics},
  volume    = {77},
  number    = {1},
  pages     = {9--22},
  year      = {2021},
  publisher = {Oxford University Press}
}


@article{williamson2023general,
  title     = {A general framework for inference on algorithm-agnostic variable importance},
  author    = {Williamson, Brian D and Gilbert, Peter B and Simon, Noah R and Carone, Marco},
  journal   = {Journal of the American Statistical Association},
  volume    = {118},
  number    = {543},
  pages     = {1645--1658},
  year      = {2023},
  publisher = {Taylor \& Francis}
}

@article{wolock2023nonparametric,
  title   = {Nonparametric variable importance for time-to-event outcomes with application to prediction of HIV infection},
  author  = {Wolock, Charles J and Gilbert, Peter B and Simon, Noah and Carone, Marco},
  journal = {arXiv preprint arXiv:2311.12726},
  year    = {2023}
}