@article{vanderweele2009distinction,
  title={On the distinction between interaction and effect modification},
  author={VanderWeele, Tyler J.},
  journal={Epidemiology},
  volume={20},
  number={6},
  pages={863--871},
  year={2009}
}

@book{rothman2008modern,
  title={{Modern Epidemiology}},
  author={Rothman, Kenneth J. and Greenland, Sander and Lash, Timothy L. and others},
  volume={3},
  year={2008},
  publisher={Wolters Kluwer Health/Lippincott Williams \& Wilkins Philadelphia}  
}

@article{lee2016exact,
  title={Exact post-selection inference, with application to the lasso},
  author={Lee, Jason D. and Sun, Dennis L. and Sun, Yuekai and Taylor, Jonathan E.},
  journal={Annals of Statistics},
  volume={44},
  number={3},
  pages={907--927},
  year={2016},
  publisher={Institute of Mathematical Statistics}
}

% MMD

@article{gretton2012kernel,
  title={A kernel two-sample test},
  author={Gretton, Arthur and Borgwardt, Karsten M. and Rasch, Malte J and Sch{\"o}lkopf, Bernhard and Smola, Alexander},
  journal={JMLR},
  volume={13},
  number={1},
  pages={723--773},
  year={2012}
}

@inproceedings{bellot2021kernel,
  title={A kernel two-sample test with selection bias},
  author={Bellot, Alexis and van der Schaar, Mihaela},
  booktitle={UAI},
  year={2021}
}

@article{muandet2021counterfactual,
  title={Counterfactual mean embeddings},
  author={Muandet, Krikamol and Kanagawa, Motonobu and Saengkyongam, Sorawit and Marukatat, Sanparith},
  journal={JMLR},
  volume={22},
  number={162},
  pages={1--71},
  year={2021}
}

@inproceedings{park2021conditional,
  title={{Conditional distributional treatment effect with kernel conditional mean embeddings and U-statistic regression}},
  author={Park, Junhyung and Shalit, Uri and Sch{\"o}lkopf, Bernhard and Muandet, Krikamol},
  booktitle={ICML},
  pages={8401--8412},
  year={2021}
}

@article{muandet2017kernel,
  title={Kernel mean embedding of distributions: {A} review and beyond},
  author={Muandet, Krikamol and Fukumizu, Kenji and Sriperumbudur, Bharath and Sch{\"o}lkopf, Bernhard and others},
  journal={Foundations and Trends{\textregistered} in Machine Learning},
  volume={10},
  number={1-2},
  pages={1--141},
  year={2017}
}

@article{sriperumbudur2010hilbert,
  title={Hilbert space embeddings and metrics on probability measures},
  author={Sriperumbudur, Bharath K. and Gretton, Arthur and Fukumizu, Kenji and Sch{\"o}lkopf, Bernhard and Lanckriet, Gert R.G.},
  journal={JMLR},
  volume={11},
  pages={1517--1561},
  year={2010}
}

@inproceedings{smola2007hilbert,
  title={A {H}ilbert space embedding for distributions},
  author={Smola, Alex and Gretton, Arthur and Song, Le and Sch{\"o}lkopf, Bernhard},
  booktitle={International Conference on Algorithmic Learning Theory},
  pages={13--31},
  year={2007}
}

% definitions 

@article{rubin1974estimating,
  title={Estimating causal effects of treatments in randomized and nonrandomized studies.},
  author={Rubin, Donald B.},
  journal={Journal of Educational Psychology},
  volume={66},
  number={5},
  pages={688},
  year={1974}
}

@article{abadie2002simple,
  title={Simple and bias-corrected matching estimators for average treatment effects},
  author={Abadie, Alberto and Imbens, Guido W.},
  journal={NBER Working Paper},
  number={t0283},
  year={2002}
}

% assumptions
@article{rosenbaum1983central,
  title={The central role of the propensity score in observational studies for causal effects},
  author={Rosenbaum, Paul R. and Rubin, Donald B.},
  journal={Biometrika},
  volume={70},
  number={1},
  pages={41--55},
  year={1983}
}

@article{elwert2014endogenous,
  title={Endogenous selection bias: {T}he problem of conditioning on a collider variable},
  author={Elwert, Felix and Winship, Christopher},
  journal={Annual Review of Sociology},
  volume={40},
  pages={31--53},
  year={2014},
  publisher={Annual Reviews}
}

% nadaraya watson

@article{nadaraya1964estimating,
  title={On estimating regression},
  author={Nadaraya, Elizbar A.},
  journal={Theory of Probability \& Its Applications},
  volume={9},
  number={1},
  pages={141--142},
  year={1964}
}

@article{watson1964smooth,
  title={Smooth regression analysis},
  author={Watson, Geoffrey S.},
  journal={Sankhy{\=a}: Indian Journal of Statistics, Series A},
  pages={359--372},
  year={1964}
}

@article{ke2019expected,
  title={Expected conditional characteristic function-based measures for testing independence},
  author={Ke, Chenlu and Yin, Xiangrong},
  journal={Journal of American Statistical Association},
  year={2019}
}

@book{silverman1986density,
  title={Density Estimation for Statistics and Data Analysis},
  author={Silverman, Bernard W.},
  volume={26},
  year={1986},
  publisher={CRC Press}
}

% propensity-based CATE estimation

@article{abrevaya2015estimating,
  title={Estimating conditional average treatment effects},
  author={Abrevaya, Jason and Hsu, Yu-Chin and Lieli, Robert P.},
  journal={Journal of Business \& Economic Statistics},
  volume={33},
  number={4},
  pages={485--505},
  year={2015}
}

@article{zhou2021ipw,
  title={{On IPW-based estimation of conditional average treatment effects}},
  author={Zhou, Niwen and Zhu, Lixing},
  journal={Journal of Statistical Planning and Inference},
  volume={215},
  pages={1--22},
  year={2021}
}

% inferring effect modifiers

@article{tukey1949comparing,
  title={Comparing individual means in the analysis of variance},
  author={Tukey, John W.},
  journal={Biometrics},
  pages={99--114},
  year={1949}
}

@article{imai2013estimating,
  title={Estimating treatment effect heterogeneity in randomized program evaluation},
  author={Imai, Kosuke and Ratkovic, Marc},
  journal={Annals of Applied Statistics},
  volume={7},
  number={1},
  pages={443--470},
  year={2013}
}

@article{tian2014simple,
author = {Lu Tian and Ash A. Alizadeh and Andrew J. Gentles and Robert Tibshirani},
title = {A simple method for estimating interactions between a treatment and a large number of covariates},
journal = {Journal of American Statistical Association},
volume = {109},
number = {508},
pages = {1517-1532},
year  = {2014}
}

@article{athey2016recursive,
  title={Recursive partitioning for heterogeneous causal effects},
  author={Athey, Susan and Imbens, Guido},
  journal={Proceedings of National Academy of Sciences},
  volume={113},
  number={27},
  pages={7353--7360},
  year={2016}
}

@article{zhao2017selective,
 author = {Zhao, Qingyuan and Small, Dylan S. and Ertefaie, Ashkan},
  title = {Selective inference for effect modification via the lasso},
  journal = {Journal of Royal Statistical Society: Series B (Statistical Methodology)},
  volume = {84},
  number = {2},
  pages = {382-413},
  year={2022}
}

@article{sechidis2021using,
  title={Using knockoffs for controlled predictive biomarker identification},
  author={Sechidis, Konstantinos and Kormaksson, Matthias and Ohlssen, David},
  journal={Statistics in Medicine},
  volume={40},
  number={25},
  pages={5453--5473},
  year={2021}
}


@article{gilad2021heterogeneous,
  title={Heterogeneous treatment effects in social networks}, 
  author={Amir Gilad and Harsh Parikh and Sudeepa Roy and Babak Salimi},
  year={2021},
  journal={arXiv preprint arXiv:2105.10591}
}

@inproceedings{hermansson2021discovering,
  title={{On discovering treatment-effect modifiers using virtual twins and causal forest ML in the presence of prognostic biomarkers}},
  author={Hermansson, Erik and Svensson, David},
  booktitle={International Conference on Computational Science and Its Applications},
  pages={624--640},
  year={2021}
}

@article{robinson1988root,
  title={Root-N-consistent semiparametric regression},
  author={Robinson, Peter M.},
  journal={Econometrica: Journal of Econometric Society},
  pages={931--954},
  year={1988}
}


% applications of effect modification discovery

@book{hernan2020causal,
  title={Causal Inference: What if. Boca Raton: Chapman \& Hill/CRC},
  author={Hern\'{a}n, Miguel A. and Robins, James M.},
  year={2020}
}

@article{lee2018discovering,
  title={Discovering effect modification in an observational study of surgical mortality at hospitals with superior nursing},
  author={Lee, Kwonsang and Small, Dylan S. and Hsu, Jesse Y. and Silber, Jeffrey H. and Rosenbaum, Paul R.},
  journal={Journal of Royal Statistical Society: Series A (Statistics in Society)},
  volume={181},
  number={2},
  pages={535--546},
  year={2018}
}

@article{schochet2014understanding,
  title={Understanding variation in treatment effects in education impact evaluations: {A}n overview of quantitative methods.},
  author={Schochet, Peter Z. and Puma, Mike and Deke, John},
  journal={National Center for Education Evaluation and Regional Assistance},
  year={2014},
  publisher={ERIC}
}

@article{taddy2016nonparametric,
  title={A nonparametric bayesian analysis of heterogenous treatment effects in digital experimentation},
  author={Taddy, Matt and Gardner, Matt and Chen, Liyun and Draper, David},
  journal={Journal of Business \& Economic Statistics},
  volume={34},
  number={4},
  pages={661--672},
  year={2016}
}

@article{jabal2021impact,
  title={{Impact of age, ethnicity, sex and prior infection status on immunogenicity following a single dose of the BNT162b2 mRNA COVID-19 vaccine: {R}eal-world evidence from healthcare workers, Israel, December 2020 to January 2021}},
  author={Jabal, Kamal Abu and Ben-Amram, Hila and Beiruti, Karine and Batheesh, Yunis and Sussan, Christian and Zarka, Salman and Edelstein, Michael},
  journal={Eurosurveillance},
  volume={26},
  number={6},
  year={2021}
}


% Recent approaches

%% Nonparameteric Bayesian models
@inproceedings{alaa2017bayesian,
  title={{Bayesian inference of individualized treatment effects using multi-task Gaussian processes}},
  author={Alaa, Ahmed M. and van der Schaar, Mihaela},
  booktitle={NeurIPS},
  pages={3427--3435},
  year={2017}
}

@article{hill2011bayesian,
  title={Bayesian nonparametric modeling for causal inference},
  author={Hill, Jennifer L.},
  journal={Journal of Computational and Graphical Statistics},
  volume={20},
  number={1},
  pages={217--240},
  year={2011}
}


%% subspace matching
@inproceedings{chang2017informative,
  title={Informative subspace learning for counterfactual inference},
  author={Chang, Yale and Dy, Jennifer},
  booktitle={AAAI},
  pages={1770--1776},
  year={2017}
}

%% causal forest
@article{wager2018estimation,
  title={Estimation and inference of heterogeneous treatment effects using random forests},
  author={Wager, Stefan and Athey, Susan},
  journal={Journal of American Statistical Association},
  volume={113},
  number={523},
  pages={1228--1242},
  year={2018}
}

@article{hahn2020bayesian,
  title={Bayesian regression tree models for causal inference: {R}egularization, confounding, and heterogeneous effects},
  author={Hahn, P. Richard and Murray, Jared S. and Carvalho, Carlos M.},
  journal={Bayesian Analysis},
  volume={15},
  number={3},
  pages={965--1056},
  year={2020}
}

%% neural representation learning
@inproceedings{johansson2016learning,
  title={Learning representations for counterfactual inference},
  author={Johansson, Fredrik and Shalit, Uri and Sontag, David},
  booktitle={ICML},
  pages={3020--3029},
  year={2016}
}

@inproceedings{shalit2017estimating,
  title={Estimating individual treatment effect: {G}eneralization bounds and algorithms},
  author={Shalit, Uri and Johansson, Fredrik D. and Sontag, David},
  booktitle={ICML},
  pages={3076--3085},
  year={2017}
}

@inproceedings{yoon2018ganite,
  title={{GANITE: {E}stimation of individualized treatment effects using generative adversarial nets}},
  author={Yoon, Jinsung and Jordon, James and Van Der Schaar, Mihaela},
  booktitle={ICLR},
  year={2018}
}

@inproceedings{hassanpour2019counterfactual,
  title={CounterFactual Regression with Importance Sampling Weights.},
  author={Hassanpour, Negar and Greiner, Russell},
  booktitle={IJCAI},
  pages={5880--5887},
  year={2019}
}

% R-learner

@article{nie2021quasi,
  title={Quasi-oracle estimation of heterogeneous treatment effects},
  author={Nie, Xinkun and Wager, Stefan},
  journal={Biometrika},
  volume={108},
  number={2},
  pages={299--319},
  year={2021}
}

% S-learner, T-learner
@article{kunzel2019metalearners,
  title={Metalearners for estimating heterogeneous treatment effects using machine learning},
  author={K{\"u}nzel, S{\"o}ren R. and Sekhon, Jasjeet S. and Bickel, Peter J. and Yu, Bin},
  journal={Proceedings of National Academy of Sciences},
  volume={116},
  number={10},
  pages={4156--4165},
  year={2019}
}



% distributional comparison

@article{kim2018causal,
  title={Causal effects based on distributional distances},
  author={Kim, Kwangho and Kim, Jisu and Kennedy, Edward H.},
  journal={arXiv preprint arXiv:1806.02935},
  year={2018}
}

% kernel methods

@inproceedings{rahimi2007random,
  title={Random Features for Large-Scale Kernel Machines.},
  author={Rahimi, Ali and Recht, Benjamin and others},
  booktitle={NeurIPS},
  volume={3},
  number={4},
  pages={5},
  year={2007}
}

@inproceedings{rahimi2008weighted,
  title={Weighted sums of random kitchen sinks: {R}eplacing minimization with randomization in learning.},
  author={Rahimi, Ali and Recht, Benjamin},
  booktitle={NeurIPS},
  pages={1313--1320},
  year={2008}
}

@book{scholkopf2002learning,
  title={Learning with kernels: {S}upport vector machines, regularization, optimization, and beyond},
  author={Sch{\"o}lkopf, Bernhard and Smola, Alexander J. and Bach, Francis and others},
  year={2002},
  publisher={MIT press}
}

% CRT

@article{liu2021fast,
    author = {Liu, Molei and Katsevich, Eugene and Janson, Lucas and Ramdas, Aaditya},
    title = "{Fast and powerful conditional randomization testing via distillation}",
    journal = {Biometrika},
    year = {2021}
}


% conditional density models

@inproceedings{sohn2015learning,
  title={Learning structured output representation using deep conditional generative models},
  author={Sohn, Kihyuk and Lee, Honglak and Yan, Xinchen},
  booktitle={NeurIPS},
  pages={3483--3491},
  year={2015}
}

@article{bashtannyk2001bandwidth,
  title={Bandwidth selection for kernel conditional density estimation},
  author={Bashtannyk, David M. and Hyndman, Rob J.},
  journal={Computational Statistics \& Data Analysis},
  volume={36},
  number={3},
  pages={279--298},
  year={2001}
}

@inproceedings{ren2016conditional,
  title={Conditional generative moment-matching networks},
  author={Ren, Yong and Zhu, Jun and Li, Jialian and Luo, Yucen},
  booktitle={NeurIPS},
  pages={2928--2936},
  year={2016}
}

@article{winkler2019learning,
  title={Learning likelihoods with conditional normalizing flows},
  author={Winkler, Christina and Worrall, Daniel and Hoogeboom, Emiel and Welling, Max},
  journal={arXiv preprint arXiv:1912.00042},
  year={2019}
}



% knockoff 

@article{candes2018panning,
  title={{Panning for gold:‘{M}odel-X’knockoffs for high dimensional controlled variable selection}},
  author={Candes, Emmanuel and Fan, Yingying and Janson, Lucas and Lv, Jinchi},
  journal={Journal of Royal Statistical Society: Series B (Statistical Methodology)},
  volume={80},
  number={3},
  pages={551--577},
  year={2018}
}

@article{barber2015controlling,
  title={Controlling the false discovery rate via knockoffs},
  author={Barber, Rina Foygel and Cand{\`e}s, Emmanuel J.},
  journal={Annals of Statistics},
  volume={43},
  number={5},
  pages={2055--2085},
  year={2015}
}

@inproceedings{jordon2018knockoffgan,
  title={{KnockoffGAN: Generating knockoffs for feature selection using generative adversarial networks}},
  author={Jordon, James and Yoon, Jinsung and van der Schaar, Mihaela},
  booktitle={ICLR},
  year={2018}
}

@article{romano2020deep,
  title={Deep knockoffs},
  author={Romano, Yaniv and Sesia, Matteo and Cand{\`e}s, Emmanuel},
  journal={Journal of American Statistical Association},
  volume={115},
  number={532},
  pages={1861--1872},
  year={2020}
}

% FDR

@article{benjamini1995controlling,
  title={Controlling the false discovery rate: {A} practical and powerful approach to multiple testing},
  author={Benjamini, Yoav and Hochberg, Yosef},
  journal={Journal of Royal statistical society: series B (Methodological)},
  volume={57},
  number={1},
  pages={289--300},
  year={1995}
}

% joint potential outcomes

@article{russell2021sharp,
  title={Sharp bounds on functionals of the joint distribution in the analysis of treatment effects},
  author={Russell, Thomas M.},
  journal={Journal of Business \& Economic Statistics},
  volume={39},
  number={2},
  pages={532--546},
  year={2021}
}

@article{chen2016inference,
  title={{Inference for the correlation coefficient between potential outcomes in the Gaussian switching regime model}},
  author={Chen, Heng and Fan, Yanqin and Liu, Ruixuan},
  journal={Journal of Econometrics},
  volume={195},
  number={2},
  pages={255--270},
  year={2016}
}

@inproceedings{shingaki2021identification,
  title={Identification and Estimation of Joint Probabilities of Potential Outcomes in Observational Studies with Covariate Information},
  author={Shingaki, Ryusei and Kuroki, Manabu},
  booktitle={NeurIPS},
  year={2021}
}

%% application of joint potential outcomes ?

@article{honore2006bounds,
  title={Bounds in competing risks models and the war on cancer},
  author={Honor{\'e}, Bo E. and Lleras-Muney, Adriana},
  journal={Econometrica},
  volume={74},
  number={6},
  pages={1675--1698},
  year={2006}
}

%% fairness-aware ML with bounds regularization

@inproceedings{chikahara2021learning,
  title={Learning individually fair classifier with path-specific causal-effect constraint},
  author={Chikahara, Yoichi and Sakaue, Shinsaku and Fujino, Akinori and Kashima, Hisashi},
  booktitle={AISTATS},
  pages={145--153},
  year={2021}
}

%% real-world datasets

@article{visser1999elevated,
  title={{Elevated C-reactive protein levels in overweight and obese adults}},
  author={Visser, Marjolein and Bouter, Lex M. and McQuillan, Geraldine M. and Wener, Mark H. and Harris, Tamara B.},
  journal={Journal of Americal Medical Association},
  volume={282},
  number={22},
  pages={2131--2135},
  year={1999}
}

@article{connors1996effectiveness,
  title={{The effectiveness of right heart catheterization in the initial care of critically III patients}},
  author={Connors, Alfred F. and Speroff, Theodore and Dawson, Neal V. and Thomas, Charles and Harrell, Frank E. and Wagner, Douglas and Desbiens, Norman and Goldman, Lee and Wu, Albert W. and Califf, Robert M. and others},
  journal={Journal of Americal Medical Association},
  volume={276},
  number={11},
  pages={889--897},
  year={1996}
}

@article{rodriguez2013obesity,
  title={Obesity and inflammation: {E}pidemiology, risk factors, and markers of inflammation},
  author={Rodr{\'\i}guez-Hern{\'a}ndez, Heriberto and Simental-Mend{\'\i}a, Luis E. and Rodr{\'\i}guez-Ram{\'\i}rez, Gabriela and Reyes-Romero, Miguel A.},
  journal={International journal of endocrinology},
  year={2013}
}

@article{olafsdottir2005c,
  title={C reactive protein levels are increased in non-allergic but not allergic asthma: {A} multicentre epidemiological study},
  author={{\'O}lafsd{\'o}ttir, Inga Sif and Gislason, Thorarinn and Thjodleifsson, B. and Olafsson, I. and Gislason, D. and J{\"o}gi, Rain and Janson, Christer},
  journal={Thorax},
  volume={60},
  number={6},
  pages={451--454},
  year={2005}
}

%% interpretabile treatment effect estimation

@article{lee2020causal,
  title={Causal rule ensemble: {I}nterpretable inference of heterogeneous treatment effects},
  author={Lee, Kwonsang and Bargagli-Stoffi, Falco J. and Dominici, Francesca},
  journal={arXiv preprint arXiv:2009.09036},
  year={2020}
}



%% for supplemental materials

@article{muller1997integral,
  title={Integral probability metrics and their generating classes of functions},
  author={M{\"u}ller, Alfred},
  journal={Advances in Applied Probability},
  volume={29},
  number={2},
  pages={429--443},
  year={1997}
}

@techreport{hoeffding1961strong,
  title={The strong law of large numbers for {U}-statistics.},
  author={Hoeffding, Wassily},
  year={1961}
}

@book{serfling2009approximation,
  title={Approximation theorems of mathematical statistics},
  author={Serfling, Robert J.},
  volume={162},
  year={2009},
  publisher={John Wiley \& Sons}
}

@article{wied2012consistency,
  title={Consistency of the kernel density estimator: {A} survey},
  author={Wied, Dominik and Wei{\ss}bach, Rafael},
  journal={Statistical Papers},
  volume={53},
  number={1},
  pages={1--21},
  year={2012}
}