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arxiv: 2605.13430 · v1 · submitted 2026-05-13 · 📊 stat.ME · cs.AI· cs.LG

Recognition: 2 theorem links

· Lean Theorem

Towards a holistic understanding of Selection Bias for Causal Effect Identification

Filip Kovacevic, Francesco Locatello, Peter Spirtes, Shimeng Huang, Yiwen Qiu

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:20 UTC · model grok-4.3

classification 📊 stat.ME cs.AIcs.LG
keywords selection biasaverage treatment effectcausal identifiabilitypropensity scoreselection probabilityobservational studiescausal inference
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The pith

Necessary and sufficient conditions identify the average treatment effect under selection bias via weak assumptions on probability classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates the identifiability of the average treatment effect when observational data suffer from selection bias, such as healthy volunteer effects in biobank studies that make the sample unrepresentative of the target population. It derives necessary and sufficient conditions for recovering the population ATE from the selected sample by characterizing the propensity score and selection probability. These conditions rely on weak assumptions about the classes of probability distributions involved. A sympathetic reader would care because selection bias is ubiquitous in real observational data and prior graphical criteria often fail to guarantee identification, so weaker conditions expand the set of recoverable causal effects.

Core claim

We provide necessary and sufficient conditions for ATE identifiability, leveraging weak assumptions on probability classes to characterize propensity score and selection probability. Compared to previous works, our results extend existing graphical identifiability criteria and offer a more comprehensive understanding of causal effect identification with strictly weaker conditions in the presence of selection bias.

What carries the argument

The necessary and sufficient identifiability conditions obtained by characterizing the propensity score and selection probability under weak assumptions on probability classes.

If this is right

  • The population ATE can be recovered from data drawn only from a selected subpopulation whenever the derived conditions are satisfied.
  • Selection bias need not preclude causal identification even when standard graphical criteria are violated.
  • Propensity-score and selection-probability characterizations become available under assumptions weaker than those required by prior graphical methods.
  • Causal-effect estimation algorithms can be built directly on the new characterizations for practical use with biased samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same probability-class approach may extend to identifiability questions involving other forms of bias such as confounding or measurement error.
  • Practical checks for the conditions could be implemented via flexible nonparametric estimators of the propensity and selection functions.
  • Study-design recommendations could follow for minimizing the severity of selection bias so that the new conditions are more likely to hold.
  • The framework may connect to existing results on transportability of causal effects across populations.

Load-bearing premise

Weak assumptions on the classes of possible probability distributions are enough to characterize the propensity score and selection probability.

What would settle it

A concrete causal structure and joint distribution in which the stated conditions hold yet the population ATE cannot be recovered from the selected sample, or in which the ATE is recoverable but the conditions fail.

Figures

Figures reproduced from arXiv: 2605.13430 by Filip Kovacevic, Francesco Locatello, Peter Spirtes, Shimeng Huang, Yiwen Qiu.

Figure 1
Figure 1. Figure 1: Illustration of selection bias in estimating the ATE of a physical activity subsidy (T) on cardiovascular health (Y ). The participation into the survey is influenced by SES, which creates a selection bias that can lead to incorrect estimates of the ATE (ATEobs ≪ ATEall) if not properly accounted for. This work aims to address the challenges posed by selection bias in causal inference. We propose a novel f… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of ATE Estimation under different noise distributions and function types, when both deterministic and non￾deterministic selection are applied. We present results for additive Gaussian and Laplace noise, and extend beyond additive noise to multiplicative noise. Overall, we observe that vanilla application of IPW leads to significantly biased estimates, while our approaches significantly improve, … view at source ↗
Figure 3
Figure 3. Figure 3: ATE estimation results on All of Us dataset. Our ap￾proach significantly decreases the bias, but not entirely. We suspect this is due to the complexity of this real-world distribution, e.g., low propensity scores (∼ 0.05), which makes the overlap very weak and leads to challenging estimation. In particular, we retrieve the last recorded BMI and T2D diagnosis status from each individual. We consider the val… view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of estimated data distributions. Finally, the conditional mean is computed as the expected value under this discrete distribution: µˆ(x) = X M j=1 yj · pˆ(yj |x) (D.17) D.3.2. GAUSSIAN MIXTURE MODEL (ANALYTICAL SOLUTION) The GMM estimates the joint density p(x, y) as a mixture of K Gaussian components with parameters πk, µk , Σk. We compute the conditional expectation analytically. Let the pa… view at source ↗
Figure 5
Figure 5. Figure 5: Additional results for Log-Log-Normal noise distribution. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
read the original abstract

Selection bias is pervasive in observational studies. For example, large scale biobanks data can exhibit ``healthy volunteer bias'' when respondents are healthier and of higher socio-economic status than the population they are meant to represent. Recovering causal effects from such sub-population is an important problem in causal inference, as estimating average treatment effects (ATE) from selected populations can result in a severely biased estimate of the ATE from the whole population. In this paper, we investigate the identifiability of the ATE under selection bias. We provide necessary and sufficient conditions for ATE identifiability, leveraging weak assumptions on probability classes to characterize propensity score and selection probability. Compared to previous works, our results extend existing graphical identifiability criteria and offer a more comprehensive understanding of causal effect identification with strictly weaker conditions in the presence of selection bias.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates identifiability of the average treatment effect (ATE) under selection bias, using examples such as healthy volunteer bias in biobanks. It claims to supply necessary and sufficient conditions for ATE identifiability by leveraging weak assumptions on probability classes that characterize the propensity score and selection probability. These conditions are asserted to extend existing graphical identifiability criteria while employing strictly weaker assumptions in the presence of selection bias.

Significance. If the claimed necessary and sufficient conditions are rigorously established, the work would advance causal inference by providing a more general characterization of ATE recovery from selected subpopulations. This could broaden the scope of identifiable causal effects beyond current graphical criteria, with direct relevance to observational data in epidemiology and social sciences where selection mechanisms are common.

major comments (1)
  1. [Abstract] Abstract: The central claim of necessary and sufficient conditions for ATE identifiability rests on 'weak assumptions on probability classes' that characterize the propensity score and selection probability, yet no explicit definitions of these probability classes, theorem statements, derivations, or counterexamples are supplied. This absence prevents verification that the conditions are indeed strictly weaker than prior graphical criteria or that they are load-bearing for identifiability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for clarity on our central claims. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim of necessary and sufficient conditions for ATE identifiability rests on 'weak assumptions on probability classes' that characterize the propensity score and selection probability, yet no explicit definitions of these probability classes, theorem statements, derivations, or counterexamples are supplied. This absence prevents verification that the conditions are indeed strictly weaker than prior graphical criteria or that they are load-bearing for identifiability.

    Authors: We agree that the abstract is a high-level summary and does not contain the full technical details, which is standard practice to maintain brevity. The explicit definitions of the probability classes (characterizing the propensity score and selection probability under our weak assumptions), the necessary and sufficient conditions, their theorem statements, derivations/proofs, and counterexamples demonstrating that the conditions are strictly weaker than prior graphical criteria are all provided in the main body of the manuscript. Specifically, these appear in Sections 3 (definitions and setup), 4 (main theorems on ATE identifiability), and 5 (comparisons to graphical criteria with counterexamples). This structure allows full verification of the claims. revision: no

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper states necessary and sufficient conditions for ATE identifiability under weak external assumptions on probability classes that characterize propensity and selection probabilities. These assumptions are introduced as independent inputs from probability theory rather than being fitted or defined in terms of the target identifiability result. No equations or steps in the provided material reduce the claimed conditions to self-referential fits, self-citations that bear the full load, or renamings of prior results. The extension of graphical criteria rests on these stated assumptions without circular reduction, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on weak assumptions about probability classes used to characterize propensity and selection probabilities; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Weak assumptions on probability classes suffice to characterize propensity score and selection probability
    Invoked to obtain necessary and sufficient conditions for ATE identifiability under selection bias.

pith-pipeline@v0.9.0 · 5453 in / 1092 out tokens · 31622 ms · 2026-05-14T19:20:22.430075+00:00 · methodology

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Lean theorems connected to this paper

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Reference graph

Works this paper leans on

85 extracted references · 32 canonical work pages · 4 internal anchors

  1. [1]

    2015 , publisher=

    Causal inference in statistics, social, and biomedical sciences , author=. 2015 , publisher=

    2015

  2. [2]

    Abouei, Amir Mohammad and Mokhtarian, Ehsan and Kiyavash, Negar and Grossglauser, Matthias , langid =. Causal

  3. [3]

    Abouei, Amir Mohammad and Mokhtarian, Ehsan and Kiyavash, Negar , year = 2024, month = jan, number =. S-. doi:10.48550/arXiv.2309.02281 , urldate =. 2309.02281 , primaryclass =

    work page doi:10.48550/arxiv.2309.02281 2024

  4. [4]

    Controlling

    Bareinboim, Elias and Pearl, Judea , year = 2011, month = aug, journal =. Controlling. doi:10.1609/aaai.v25i1.8056 , urldate =

    work page doi:10.1609/aaai.v25i1.8056 2011

  5. [5]

    Bellot, Alexis , langid =. Towards

  6. [6]

    Managing Extreme

    Bengio, Yoshua and Hinton, Geoffrey and Yao, Andrew and Song, Dawn and Abbeel, Pieter and Darrell, Trevor and Harari, Yuval Noah and Zhang, Ya-Qin and Xue, Lan and. Managing Extreme. Science , volume =. doi:10.1126/science.adn0117 , urldate =. 2310.17688 , primaryclass =

    work page doi:10.1126/science.adn0117

  7. [7]

    , author=

    Constructing Separators and Adjustment Sets in Ancestral Graphs. , author=. CI@ UAI , pages=

  8. [8]

    Cai, Yang and Kalavasis, Alkis and Mamali, Katerina and Mehrotra, Anay and Zampetakis, Manolis , year = 2025, publisher =. What. doi:10.48550/ARXIV.2506.04194 , urldate =

    work page doi:10.48550/arxiv.2506.04194 2025

  9. [9]

    Generalized

    Correa, Juan and Tian, Jin and Bareinboim, Elias , year = 2018, month = apr, journal =. Generalized. doi:10.1609/aaai.v32i1.12125 , urldate =

    work page doi:10.1609/aaai.v32i1.12125 2018

  10. [10]

    and Tian, Jin and Bareinboim, Elias , year = 2019, month = jul, journal =

    Correa, Juan D. and Tian, Jin and Bareinboim, Elias , year = 2019, month = jul, journal =. Identification of. doi:10.1609/aaai.v33i01.33012744 , urldate =

    work page doi:10.1609/aaai.v33i01.33012744 2019

  11. [11]

    Detecting

    Cummings, Jesse and Snorrason, El. Detecting

  12. [12]

    Dai, Haoyue and Ng, Ignavier and Sun, Jianle and Tang, Zeyu and Luo, Gongxu and Dong, Xinshuai and Spirtes, Peter and Zhang, Kun , year = 2025, month = mar, number =. When. doi:10.48550/arXiv.2503.07302 , urldate =. 2503.07302 , primaryclass =

    work page doi:10.48550/arxiv.2503.07302 2025

  13. [13]

    Daskalakis, Constantinos and Kontonis, Vasilis and Tzamos, Christos and Zampetakis, Manolis , year = 2021, month = jun, number =. A. doi:10.48550/arXiv.2106.15908 , urldate =. 2106.15908 , primaryclass =

    work page doi:10.48550/arxiv.2106.15908 2021

  14. [14]

    Endogenous

    Elwert, Felix and Winship, Christopher , year = 2014, month = jul, journal =. Endogenous. doi:10.1146/annurev-soc-071913-043455 , urldate =

    work page doi:10.1146/annurev-soc-071913-043455 2014

  15. [15]

    Review of

    Glymour, Clark and Zhang, Kun and Spirtes, Peter , year = 2019, journal =. Review of

    2019

  16. [16]

    Quantifying

    Greenland, Sander , year = 2003, month = may, journal =. Quantifying. doi:10.1097/01.EDE.0000042804.12056.6C , urldate =

    work page doi:10.1097/01.ede.0000042804.12056.6c 2003

  17. [17]

    International Conference on Machine Learning , pages=

    Causal identification under markov equivalence: Completeness results , author=. International Conference on Machine Learning , pages=. 2019 , organization=

    2019

  18. [18]

    Jaber, Amin and Zhang, Jiji and Bareinboim, Elias , year = 2018, month = jul, pages =. A. Proceedings of the. doi:10.24963/ijcai.2018/697 , urldate =

    work page doi:10.24963/ijcai.2018/697 2018

  19. [19]

    Estimating

    Kalisch, Markus , langid =. Estimating

  20. [20]

    Knox, Dean and Lowe, Will and Mummolo, Jonathan , year = 2020, journal =. Can. doi:10.2139/ssrn.3940802 , urldate =

    work page doi:10.2139/ssrn.3940802 2020

  21. [21]

    Kummerfeld, Erich and Ramsey, Joseph , year = 2016, month = aug, pages =. Causal. Proceedings of the 22nd. doi:10.1145/2939672.2939838 , urldate =

    work page doi:10.1145/2939672.2939838 2016

  22. [22]

    Ancestral Causal Inference

    Magliacane, Sara and Claassen, Tom and Mooij, Joris M. , year = 2017, month = jan, number =. Ancestral. doi:10.48550/arXiv.1606.07035 , urldate =. 1606.07035 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1606.07035 2017

  23. [23]

    Am J Epidemiol , volume =

    Simple Graphical Rules for Assessing Selection Bias in General-Population and Selected-Sample Treatment Effects , author =. Am J Epidemiol , volume =. doi:10.1093/aje/kwae145 , urldate =

    work page doi:10.1093/aje/kwae145

  24. [24]

    and Eulig, Elias and Noceti, Nicoletta and Rosasco, Lorenzo and Janzing, Dominik and Aragam, Bryon and Locatello, Francesco , year = 2023, month = nov, urldate =

    Montagna, Francesco and Mastakouri, Atalanti A. and Eulig, Elias and Noceti, Nicoletta and Rosasco, Lorenzo and Janzing, Dominik and Aragam, Bryon and Locatello, Francesco , year = 2023, month = nov, urldate =. Assumption Violations in Causal Discovery and the Robustness of Score Matching , booktitle =

    2023

  25. [25]

    and Baker, Leanne F

    Mudge, Joseph F. and Baker, Leanne F. and Edge, Christopher B. and Houlahan, Jeff E. , editor =. Setting an. PLoS ONE , volume =. doi:10.1371/journal.pone.0032734 , urldate =

    work page doi:10.1371/journal.pone.0032734

  26. [26]

    Ogarrio, Juan Miguel and Spirtes, Peter and Ramsey, Joe , langid =. A

  27. [27]

    Ancestral Graph

    Richardson, Thomas and Spirtes, Peter , year = 2002, month = aug, journal =. Ancestral Graph. doi:10.1214/aos/1031689015 , urldate =

    work page doi:10.1214/aos/1031689015 2002

  28. [28]

    Richardson, Thomas S and Robins, James M , langid =. Single

  29. [29]

    and Nisimov, Shami and Gurwicz, Yaniv and Novik, Gal , year = 2022, month = jan, number =

    Rohekar, Raanan Y. and Nisimov, Shami and Gurwicz, Yaniv and Novik, Gal , year = 2022, month = jan, number =. Iterative. arXiv , langid =:2111.04095 , primaryclass =

    work page arXiv 2022

  30. [30]

    Causal Inference in the Presence of Latent Variables and Selection Bias

    Spirtes, Peter L. and Meek, Christopher and Richardson, Thomas S. , year = 2013, month = feb, number =. Causal. doi:10.48550/arXiv.1302.4983 , urldate =. 1302.4983 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1302.4983 2013

  31. [31]

    Spirtes, Peter and Zhang, Jiji , year = 2014, month = nov, journal =. A. doi:10.1214/13-STS429 , urldate =. arXiv , langid =:1502.00829 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1214/13-sts429 2014

  32. [32]

    , year = 2012, month = jul, journal =

    Swanson, James M. , year = 2012, month = jul, journal =. The. doi:10.1016/S0140-6736(12)61179-9 , urldate =

    work page doi:10.1016/s0140-6736(12)61179-9 2012

  33. [33]

    Tian, Jin and Pearl, Judea , langid =. A

  34. [34]

    Geometry of the Faithfulness Assumption in Causal Inference , author =. Ann. Statist. , volume =. doi:10.1214/12-AOS1080 , urldate =. arXiv , langid =:1207.0547 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1214/12-aos1080

  35. [36]

    Reweighting

    Van Alten, Sjoerd and Domingue, Benjamin W and Faul, Jessica and Galama, Titus and Marees, Andries T , year = 2024, month = apr, journal =. Reweighting. doi:10.1093/ije/dyae054 , urldate =

    work page doi:10.1093/ije/dyae054 2024

  36. [37]

    Versteeg, Philip and Mooij, Joris and Zhang, Cheng , year = 2022, month = jun, pages =. Local. Proceedings of the

    2022

  37. [38]

    How and Why Alpha Should Depend on Sample Size:

    Wulff, Jesper N and Taylor, Luke , year = 2023, month = nov, journal =. How and Why Alpha Should Depend on Sample Size:. doi:10.1177/14761270231214429 , urldate =

    work page doi:10.1177/14761270231214429 2023

  38. [39]

    Zhang, Jiji , langid =. Causal

  39. [40]

    Zhang, Jiji , year = 2013, month = jun, journal =. A. doi:10.1093/bjps/axs005 , urldate =

    work page doi:10.1093/bjps/axs005 2013

  40. [41]

    Artificial Intelligence , volume =

    On the Completeness of Orientation Rules for Causal Discovery in the Presence of Latent Confounders and Selection Bias , author =. Artificial Intelligence , volume =. doi:10.1016/j.artint.2008.08.001 , urldate =

    work page doi:10.1016/j.artint.2008.08.001 2008

  41. [42]

    Zhang, Yichi and Lu, Haidong , year = 2025, month = dec, number =. On the. doi:10.48550/arXiv.2502.00924 , urldate =. 2502.00924 , primaryclass =

    work page doi:10.48550/arxiv.2502.00924 2025

  42. [43]

    , author=

    On the Identifiability and Estimation of Functional Causal Models in the Presence of Outcome-Dependent Selection. , author=. UAI , year=

  43. [44]

    Detecting and

    Zheng, Yujia and Tang, Zeyu and Qiu, Yiwen and Sch. Detecting and. doi:10.48550/arXiv.2407.00529 , urldate =. 2407.00529 , primaryclass =

    work page doi:10.48550/arxiv.2407.00529

  44. [45]

    Current Epidemiology Reports , volume=

    Selection mechanisms and their consequences: understanding and addressing selection bias , author=. Current Epidemiology Reports , volume=. 2020 , publisher=

    2020

  45. [46]

    arXiv preprint arXiv:2512.11219 , year=

    Latent variable causal discovery under selection bias , author=. arXiv preprint arXiv:2512.11219 , year=

    work page arXiv

  46. [47]

    Econometrica: Journal of the econometric society , pages=

    Sample selection bias as a specification error , author=. Econometrica: Journal of the econometric society , pages=. 1979 , publisher=

    1979

  47. [48]

    Uncertainty in Artificial Intelligence , pages=

    Causal calculus in the presence of cycles, latent confounders and selection bias , author=. Uncertainty in Artificial Intelligence , pages=. 2020 , organization=

    2020

  48. [49]

    Conference on Causal Learning and Reasoning , pages=

    Local constraint-based causal discovery under selection bias , author=. Conference on Causal Learning and Reasoning , pages=. 2022 , organization=

    2022

  49. [50]

    arXiv preprint arXiv:2401.06925 , year=

    Modeling Latent Selection with Structural Causal Models , author=. arXiv preprint arXiv:2401.06925 , year=

    work page arXiv

  50. [51]

    Epidemiology , pages=

    A structural approach to selection bias , author=. Epidemiology , pages=. 2004 , publisher=

    2004

  51. [52]

    Artificial Intelligence , volume=

    On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias , author=. Artificial Intelligence , volume=. 2008 , publisher=

    2008

  52. [53]

    Current Psychology , volume=

    A population-based investigation of participation rate and self-selection bias in momentary data capture and survey studies , author=. Current Psychology , volume=. 2024 , publisher=

    2024

  53. [54]

    2001 , publisher=

    Causation, prediction, and search , author=. 2001 , publisher=

    2001

  54. [55]

    2009 , publisher=

    Causality , author=. 2009 , publisher=

    2009

  55. [56]

    arXiv preprint arXiv:2506.04194 , year=

    What Makes Treatment Effects Identifiable? Characterizations and Estimators Beyond Unconfoundedness , author=. arXiv preprint arXiv:2506.04194 , year=

    work page arXiv

  56. [57]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Causal effect identification by adjustment under confounding and selection biases , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  57. [58]

    , author=

    Estimation of non-normalized statistical models by score matching. , author=. Journal of Machine Learning Research , volume=

  58. [59]

    Biometrika , volume=

    The central role of the propensity score in observational studies for causal effects , author=. Biometrika , volume=. 1983 , publisher=

    1983

  59. [60]

    American Journal of Epidemiology , volume=

    Simple graphical rules for assessing selection bias in general-population and selected-sample treatment effects , author=. American Journal of Epidemiology , volume=. 2025 , publisher=

    2025

  60. [61]

    2010 , publisher=

    Causal inference , author=. 2010 , publisher=

    2010

  61. [62]

    Epidemiology , volume=

    Toward a clearer definition of selection bias when estimating causal effects , author=. Epidemiology , volume=. 2022 , publisher=

    2022

  62. [63]

    International Conference on Machine Learning , pages=

    Budgeted experiment design for causal structure learning , author=. International Conference on Machine Learning , pages=. 2018 , organization=

    2018

  63. [64]

    International Conference on Machine Learning , pages=

    Cost-optimal learning of causal graphs , author=. International Conference on Machine Learning , pages=. 2017 , organization=

    2017

  64. [65]

    The Journal of Machine Learning Research , volume=

    Experiment selection for causal discovery , author=. The Journal of Machine Learning Research , volume=

  65. [66]

    Journal of Machine Learning Research , volume=

    Active learning of causal networks with intervention experiments and optimal designs , author=. Journal of Machine Learning Research , volume=

  66. [67]

    Artificial Intelligence and Statistics , pages=

    Controlling selection bias in causal inference , author=. Artificial Intelligence and Statistics , pages=. 2012 , organization=

    2012

  67. [68]

    Probabilistic and causal inference: The works of Judea Pearl , pages=

    Recovering from selection bias in causal and statistical inference , author=. Probabilistic and causal inference: The works of Judea Pearl , pages=

  68. [69]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    Recovering causal effects from selection bias , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  69. [70]

    1977 , publisher=

    Sample selection bias as a specification error (with an application to the estimation of labor supply functions) , author=. 1977 , publisher=

    1977

  70. [71]

    Epidemiology , volume=

    Marginal structural models and causal inference in epidemiology , author=. Epidemiology , volume=

  71. [72]

    2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) , pages=

    Efficient truncated statistics with unknown truncation , author=. 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) , pages=. 2019 , organization=

    2019

  72. [73]

    The Annals of mathematical statistics , pages=

    Estimating the mean and variance of normal populations from singly truncated and doubly truncated samples , author=. The Annals of mathematical statistics , pages=. 1950 , publisher=

    1950

  73. [74]

    2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) , pages=

    Efficient Statistics With Unknown Truncation, Polynomial Time Algorithms, Beyond Gaussians , author=. 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) , pages=. 2024 , organization=

    2024

  74. [75]

    Econometrica , volume=

    A conditional likelihood ratio test for structural models , author=. Econometrica , volume=. 2003 , publisher=

    2003

  75. [76]

    PLoS genetics , volume=

    Genetic sensitivity analysis: Adjusting for genetic confounding in epidemiological associations , author=. PLoS genetics , volume=. 2021 , publisher=

    2021

  76. [77]

    Nature Genetics , volume=

    Adjusting for genetic confounders in transcriptome-wide association studies improves discovery of risk genes of complex traits , author=. Nature Genetics , volume=. 2024 , publisher=

    2024

  77. [78]

    2018 , publisher=

    Machine learning: a practical approach on the statistical learning theory , author=. 2018 , publisher=

    2018

  78. [79]

    Annals of the Institute of Statistical Mathematics , volume=

    New approaches to statistical learning theory , author=. Annals of the Institute of Statistical Mathematics , volume=. 2003 , publisher=

    2003

  79. [80]

    IEEE transactions on neural networks , volume=

    An overview of statistical learning theory , author=. IEEE transactions on neural networks , volume=. 1999 , publisher=

    1999

  80. [81]

    2004 , publisher=

    The elements of statistical learning: data mining, inference, and prediction , author=. 2004 , publisher=

    2004

Showing first 80 references.