Pith. sign in

REVIEW 3 major objections 3 minor 1 cited by

Partial identification via conditional linear programs: estimation and policy learning

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two debiased estimators make partial-identification bounds asymptotically normal at root-n and support Wald confidence intervals without vertex enumeration.

desk verdict A useful unification and a genuinely practical BFS estimator, but the entropic estimator's root-n theory has a real gap for small margin exponents. read the letter →

arxiv 2506.12215 v2 pith:RLRVDSNZ submitted 2025-06-13 stat.ME econ.EM

Eli Ben-Michael This is my paper · ORCID
classification stat.MEecon.EM MSC 62G0562G2090C05
keywords partialidentificationconditionallinearprogramsdebiasedestimationentropicregularizationpolicylearningmarginconditionasymptoticnormalitycausalbounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops a way to estimate, test, and learn policies for parameters that are only partially identified, in the common case where bounds on the parameter can be written as expectations of solutions to linear programs whose objectives and constraints depend on covariates. It proposes two debiased estimators: a plug-in basic-feasible-solution estimator that reads de-biasing information out of an ordinary LP solver, and an entropic-regularized estimator that smooths the program so that solutions are differentiable in the nuisance functions. The paper's central theoretical claim is that both estimators are asymptotically normal at the $n^{1/2}$ rate, and therefore support Wald confidence intervals for the identified set, whenever nuisance estimates converge fast enough and a margin condition on sub-optimality gaps holds. It also shows that optimizing these estimated bounds yields policies whose excess regret is controlled by the policy class's Rademacher complexity plus the same debiased error rates. The methods are applied to the Oregon Medicaid experiment to bound counterfactual and regret estimands for emergency-department visits.

What carries the argument

The central object is the conditional linear program $\theta_L=\mathbb{E}[\min_{p\in P(X)}\langle c(X),p\rangle]$ with feasible set $P(x)=\{p\in\mathbb{R}_+^K:Ap=b(x)\}$; the paper rewrites the program over basic feasible solutions $p=A_B^{-1}b(x)$. The plug-in estimator selects bases $B$ by optimizing with estimated $\hat b,\hat c$ and debiases using Riesz-representer corrections $\hat\varphi^{(b)},\hat\varphi^{(c)}$---weighted residual functions that make debiased estimates first-order robust to nuisance error. The entropic estimator replaces the objective with $\langle c,p\rangle+(1/\eta)\sum_k p_k(\log p_k-1)$, whose dual variables give an explicit smooth solution and Jacobians $\nabla_b p^\eta,\nabla_c p^\eta$ used in the debiasing Taylor expansion. The argument is carried by the sub-optimality gap $\Delta_L(x)$ and the margin condition $P(0<\Delta_L(X)\le t)\lesssim t^\alpha$, which controls the probability that plug-in bases are misclassified, and by the Rademacher complexity of the policy class for the policy-learning results.

What would settle it

Simulate a data-generating process with two nearly identical feasible vertices, so that $P(0<\Delta_L(X)\le t)$ decays like $t^{0.1}$; compute the plug-in BFS estimator's bias under nuisance error $\varepsilon$ and check whether it decreases like $\varepsilon^{1.1}$ and whether 95% Wald intervals cover the true lower bound. If coverage drops well below nominal as $n$ grows, the margin-dependent expansion in Theorem 1 is falsified.

Watch

Extended reading notes

Core claim

The central claim is that plug-in and entropic-regularized estimates of bounds defined by conditional linear programs can be debiased so that inference and policy learning proceed as if the bounds were smooth point-identified functionals. Theorem 1 shows the plug-in BFS estimator satisfies $\hat\theta_L-\theta_L$ equals a mean-zero expansion plus $O_p((\|\hat b-b\|_\infty+\|\hat c-c\|_\infty)^{1+\alpha}+r_n+\|\hat b-b\|_2\|\hat c-c\|_2)+o_p(n^{-1/2})$; Theorem 2 shows the entropic estimator satisfies an analogous expansion with an added $e^{-\eta}$ approximation-error term. Corollaries give conditions---unique optimal bases, non-degenerate solutions, or zero conditional variance of the debiasing functions---under which the asymptotic variance depends only on true optimal bases, and the empirical variance estimators yield Wald intervals. Theorems 3 and 4 bound excess regret for estimated policies by $R_n(\Pi)+\|\hat b-b\|_\infty^{1+\alpha}+r_n$ and, for entropic policies, by $\eta R_n(\Pi)+e^{-\eta}+r_n+\eta^2\|\hat b-b\|_2^2$. The paper applies the estimators to the Oregon health insurance experiment, estimating bounds on extra emergency-department visits under waitlist randomization versus an oracle ED-minimizing rule, and on regret under power-law collective utility functions.

Load-bearing premise

The central claim collapses if the margin condition fails---that is, if the gap between the best and next-best feasible solutions is frequently tiny---because then the basis-misclassification bias decays slowly; the entropic result additionally requires an unverifiable unit-level lower bound on the regularization parameter.

Editorial extensions

If this is right

  • Analysts can build Wald confidence intervals for partially identified parameters using standard LP solver output, without enumerating the possibly combinatorial set of feasible vertices.
  • Flexible machine-learning estimates of constraint and objective functions are allowed: only their debiased rate and margin-adjusted errors need to vanish faster than $n^{-1/2}$, not the raw nuisance errors themselves.
  • The entropic regularization level $\eta$ acts as a sensitivity parameter moving continuously between the no-assumption bounds ($\eta\to\infty$) and a maximum-entropy point identification ($\eta\to 0$), with approximation error decaying like $e^{-\eta}$.
  • For policy learning, excess regret of the estimated optimal policy is controlled by the Rademacher complexity of the policy class plus the debiased estimation error, so choosing a simpler policy class can compensate for slower nuisance convergence.
  • In point-identified cases such as the average treatment effect on the treated, the estimators reduce to standard debiased estimators and are insensitive to the regularization choice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The margin exponent $\alpha$ is in practice unknown; a natural sensitivity analysis would re-estimate the bounds under worst-case small $\alpha$ or replace normality-based intervals with bounds that do not rely on the margin condition.
  • Because Theorem 2's condition $\eta\ge (R_1(X_i)+R_H(X_i))/\Delta_L(X_i)$ depends on unobserved sub-optimality gaps, a testable extension would estimate per-unit gaps and choose $\eta$ from their empirical distribution, then compare coverage across choices.
  • The dual formulation of the entropic program suggests an immediate extension to continuous outcomes: discretize the outcome space, compute the dual variables, and check whether the resulting bounds remain valid as the grid refines, which the paper flags as an open limitation.
  • The excess-regret rates imply an explicit trade-off: shrinking the policy class reduces the Rademacher term but may increase the approximation error from not containing the true optimum; a data-driven rule for model class selection could be derived from Theorem 4's bound.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This paper develops a unified framework for estimating and drawing inference about partially identified parameters whose sharp bounds are expectations of covariate-conditional linear programs. The estimand class includes functions of the joint distribution of potential outcomes, inequality-aware collective utility functions, and instrumental-variable settings. The paper proposes two debiased estimators: a plug-in basic feasible solution (BFS) estimator and an entropic-regularized estimator, and states asymptotic normality results for both (Theorems 1 and 2, Corollaries 1--2 and D.1) as well as excess-regret bounds for policy learning (Theorems 3 and 4). The methods are illustrated with simulations and an Oregon Medicaid application. The BFS estimator is computationally attractive because it avoids vertex enumeration, and the paper contains detailed appendix proofs and extensive simulation comparisons.

Significance. If the main theorems hold, the paper provides a substantial unification: it handles several important partial-identification problems with a single estimation and inference recipe, and it shows that debiasing can be done using only the output of standard LP solvers or entropic dual solutions, without closed-form bound derivations. The detailed appendix proofs, the explicit treatment of nuisance estimation via Riesz representers, and the simulation study comparing against log-sum-exp approximations are clear strengths. The main weakness is that the entropic estimator's root-n normality is only established under a pointwise, unverifiable condition on the regularization parameter η whose compatibility with the stated margin condition is much more restricted than the paper acknowledges; as a result, the claim that both estimators are asymptotically normal is not fully supported over the parameter space permitted by Assumption 3.

major comments (3)
  1. [Theorem 2] Theorem 2 conditions on the realized sample: it requires η ≥ (R1(Xi)+RH(Xi))/Δ_L(Xi) for every observed Xi with Δ_L(Xi) > 0. Under Assumption 3, the smallest positive sub-optimality gap in an iid sample has order n^{-1/α} under a Pareto-tail model that saturates the margin condition, so this condition forces η to grow at least like n^{1/α} with probability tending to one. The same theorem also requires n^{-1/4}η → 0. These requirements are simultaneously satisfiable only if n^{1/α} = o(n^{1/4}), i.e., α > 4, whereas Assumption 3 only states α > 0. The discussion in Section 3.2 acknowledges that 'the minimal size of η grows too quickly' may occur in some data generating processes, but the theorem as stated claims normality under Assumptions 1--3 without this restriction. The statement needs either an explicit lower-tail condition on Δ_L, a restricted range for α, or a different asymptotic regime for the entropic estimator.
  2. [Appendix D.1, Corollary D.1] Corollary D.1 removes the pointwise η condition but requires n^{1/2}η^{-(1+α)} → 0 and n^{-1/4}η → 0. Writing η = n^β gives β > 1/(2(1+α)) and β < 1/4. For α ≤ 1 these inequalities cannot both hold, so no admissible η sequence exists. For 1 < α ≤ 4 the admissible interval is nonempty only for β near 1/4, while the minimal η forced by the pointwise condition (n^{1/α}) is incompatible with β < 1/4 unless α > 4. Thus for a substantial part of the parameter space permitted by Assumption 3, neither Theorem 2 nor Corollary D.1 supplies a choice of η under which the entropic estimator is root-n normal. The same issue affects the second claim in Theorem 4, which requires the pointwise η condition uniformly over all π ∈ Π.
  3. [Section 5, Medicaid enrollment analysis] The empirical section reports that for 14% of units the linear program was infeasible and states 'We exclude these units from the analysis.' This exclusion changes the estimand: the reported bounds and confidence intervals are for the subpopulation for which the estimated constraints are feasible, not for the Oregon sample described in the paper. Since feasibility depends on estimated nuisance functions, this is a form of selection on estimated values and should be addressed explicitly, for example by reporting sensitivity analyses, re-weighting the feasible units, or estimating bounds under a model for the infeasible units.
minor comments (3)
  1. [Lemma D.3 proof] In the proof of Lemma D.3, the Jacobians for the upper-bound dual problem are labeled with subscript L (∇_b λ^η_L and ∇_c λ^η_L) after introducing G_U; these should be subscript U. This is a typographical error, but it makes the proof harder to follow.
  2. [Section 2.1, notation] The notation ∥f∥_∞ for vector-valued functions is defined as max_i sup_x |f_i(x)|, but this is not stated until after the first use; a one-sentence definition earlier would improve readability.
  3. [Section 3.2] The paragraph following Theorem 2 says one can 'expect to find an η that is larger than this minimal value for a finite sample,' but this is not a mathematical guarantee and is precisely the point where the compatibility problem with Assumption 3 arises; the text should flag this as a condition that must be verified or imposed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the debiased BFS and entropic estimators are derived from influence functions, Riesz representers, and external approximation-error bounds, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained and does not reduce to its own inputs. The target lower and upper bounds are defined by conditional linear programs in Equation (2), and the estimators in Equations (6) and (10) are constructed directly from plugin LP solutions plus debiasing terms built from Assumption 1's debiasing functions, which are standard efficient-influence-function/Riesz-representer objects. Theorem 1 and Theorem 2 give asymptotic expansions whose bias terms are explicitly bounded in terms of nuisance-error rates, the margin parameter, and the regularization parameter; no term is a fitted quantity that is then relabeled a prediction. The margin condition in Assumption 3 is an exogenous regularity condition, not an output of the estimation procedure. The entropic approximation-error bounds are cited to Weed (2018), an external, independently stated result, and the proof of Theorem 2 relies on that external bound rather than on a self-citation. The paper does cite the author's own prior work in the related-literature discussion and in the policy-learning examples, but those citations are contextual and not load-bearing assumptions for the main theorems. The application uses the external Oregon Health Insurance Experiment data. The appendix even includes an honest limitation: the authors note that 'depending on the tail behavior of the sub-optimality gaps Delta_L(X_i), it may be possible that under some data generating processes the minimal size of eta grows too quickly,' and they provide a fallback result, Corollary D.1, for that case. This is a genuine technical limitation about the scope of Theorem 2, not a circular step: it does not make the claimed normality equivalent to an assumption or to a self-cited uniqueness result. Similarly, the skeptic's concern about the compatibility of the pointwise eta condition with Assumption 3 for small alpha is a correctness and assumption-compatibility issue, not a circularity issue, because the theorem's conclusion is not forced by fitting or by definitional identity. Overall, the central claims have independent statistical content and are derived from external semiparametric theory and LP/entropic-regularization results.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central methodology rests on debiasing function existence, sup-norm consistency with a de-biased rate, and margin conditions on the sub-optimality gap. The margin condition is the most fragile because it is unverifiable and directly controls the convergence rate. The empirical application adds standard causal assumptions plus a post-hoc exclusion of infeasible units.

free parameters (2)
  • Entropic regularization hyperparameter eta = eta = 100 in the Medicaid regret analysis; eta^{-1} ~ n^{-1/2} and ~1/log(n) in simulations
    Chosen by hand as a sensitivity parameter. It controls the tradeoff between approximation error and estimation stability; results vary with eta, and Theorem 2 requires eta to grow with n while satisfying a data-dependent lower bound.
  • Margin parameter alpha = Unknown; not estimated
    The margin condition assumes P(0 < Delta_L(X) <= t) ~ t^alpha. The rates in Theorems 1 and 3 depend on alpha, but alpha is not estimated or verified in applications.
assumptions (8)
  • domain assumption Feasible set P(x) is nonempty and bounded for all x, and A has full row rank equal to J.
    Section 2.1 assumes P(x) is nonempty and bounded for all x, and that A is pre-processed to remove redundant constraints. This ensures the LP has basic feasible solutions.
  • domain assumption Assumption 1: debiasing functions with conditional mean zero exist for b and c.
    Section 3 introduces de-biasing functions phi^(b) and phi^(c) satisfying E[phi | X] = 0 and an unbiasedness property for any bounded plug-in function. This is the foundation for the debiased estimators.
  • domain assumption Assumption 2: sup-norm consistency of nuisance and debiasing functions, plus a de-biased rate r_n for linear functionals.
    Section 3 requires sup-norm consistency and a rate r_n controlling the bias of de-biased estimates of linear functions of b and c. This rate appears in all main theorems.
  • domain assumption Assumption 3: margin condition P(0 < Delta_L(X) <= t) ~ t^alpha.
    Equation (7) defines the sub-optimality gap and Assumption 3 controls its tail near zero. The plug-in basis misclassification bias decays as (nuisance error)^(1+alpha), so alpha is load-bearing for root-n inference.
  • domain assumption Assumption 4: boundedness of b, c, phi, and Lipschitz continuity of c in the policy pi.
    Section 4 imposes boundedness and Lipschitz assumptions to apply empirical process results and Talagrand's contraction principle in the policy learning theorems.
  • domain assumption Assumption 5: strong margin condition uniformly over policies in Pi.
    Section 4.1 requires the margin condition to hold uniformly over the policy class Pi to control the sup-norm bias of the estimated bound in Theorem 3.
  • standard math Weed (2018) exponential bound on the entropic approximation error.
    Theorem 2 and Corollary D.1 rely on Weed's explicit analysis of the entropic penalty in linear programming to bound the approximation error of the regularized bounds.
  • domain assumption Strong ignorability of the instrument and exclusion restriction in the Oregon application.
    Section 5 assumes that being chosen off the waitlist is ignorable and that the instrument affects the outcome only through Medicaid enrollment. These are standard but untestable assumptions for the empirical claims.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Partial identification via conditional linear programs: estimation and policy learning." pith.science (2026). https://pith.science/paper/RLRVDSNZ

@misc{pith2026250612215,
  author       = {Pith},
  title        = {Pith review of: Partial identification via conditional linear programs: estimation and policy learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLRVDSNZ}},
  note         = {Machine review of arXiv:2506.12215}
}
read the original abstract

Many important quantities of interest are only partially identified from observable data: the data can limit them to a set of plausible values, but not uniquely determine them. This paper develops a unified framework for covariate-assisted estimation, inference, and decision making in partial identification problems where the parameter of interest satisfies a series of linear constraints, conditional on covariates. In such settings, bounds on the parameter can be written as expectations of solutions to conditional linear programs that optimize a linear function subject to linear constraints, where both the objective function and the constraints may depend on covariates and need to be estimated from data. Examples include estimands involving the joint distributions of potential outcomes, policy learning with inequality-aware value functions, and instrumental variable settings. We propose two de-biased estimators for bounds defined by conditional linear programs. The first directly solves the conditional linear programs with plugin estimates and uses output from standard LP solvers to de-bias the plugin estimate, avoiding the need for computationally demanding vertex enumeration of all possible solutions for symbolic bounds. The second uses entropic regularization to create smooth approximations to the conditional linear programs, trading a small amount of approximation error for improved estimation and computational efficiency. We establish conditions for asymptotic normality of both estimators, show that both estimators are robust to first-order errors in estimating the conditional constraints and objectives, and construct Wald-type confidence intervals for the partially identified parameters. These results also extend to policy learning problems where the value of a decision policy is only partially identified. We apply our methods to a study on the effects of Medicaid enrollment.

Figures

Figures reproduced from arXiv: 2506.12215 by the authors.

Figure 1
Figure 1. Estimated bounds on the expected number of additional ED visits and the per [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. (a) Estimated bounds on the regret of the randomization policy versus the ED [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Bounds on the expected number of additional ED visits and the percentage of [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Semiparametric inference on identification sets in choice modeling

    math.ST 2026-07 accept novelty 6.0 of 10

    New inference theory for the identified interval of counterfactual choice probabilities when the mixing distribution is unrestricted: LP duality, von Mises expansions, and an EM-based membership certificate.

Reference graph

Works this paper leans on

50 extracted references · 41 canonical work pages · cited by 1 Pith paper

  1. [1]

    Audibert, J. Y. and Tsybakov, A. B. (2007). Fast learning rates for plug-in classifiers . Annals of Statistics , 35(2):608--633

    work page 2007

  2. [2]

    and Pearl, J

    Balke, A. and Pearl, J. (1994). Counterfactual Probabilities: Computational Methods, Bounds and Applications . In Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence , pages 46--54

    work page 1994

  3. [3]

    and Pearl, J

    Balke, A. and Pearl, J. (1997). Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association , 92(439):1171--1176

    1997

  4. [4]

    J., Huang, M., Imai, K., Jiang, Z., and Shin, S

    Ben-Michael, E., Greiner, D. J., Huang, M., Imai, K., Jiang, Z., and Shin, S. (2024a). Does AI help humans make better decisions? A statistical evaluation framework for experimental and observational studies. arXiv:2403.12108

    arXiv 2024

  5. [5]

    J., Imai, K., and Jiang, Z

    Ben-Michael, E., Greiner, D. J., Imai, K., and Jiang, Z. (2021). Safe Policy Learning through Extrapolation: Application to Pre-trial Risk Assessment

    work page 2021

  6. [6]

    Ben-Michael, E., Imai, K., and Jiang, Z. (2024b). Policy Learning with Asymmetric Counterfactual Utilities . Journal of the American Statistical Association , 119(548):3045--3058

    work page 2024

  7. [7]

    Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal , 21(1):C1--C68

    2018

  8. [8]

    Cui, Y. (2021). Individualized decision making under partial identification: three perspectives, two optimality results, and one paradox . Harvard Data Science Review . Just accepted

    work page 2021

Show all 50 references
  1. [9]

    Cuturi, M. (2013). Sinkhorn Distances : Lightspeed Computation of Optimal Transport . In Advances in Neural Information Processing Systems , volume 26

    2013

  2. [10]

    D'Adamo, R. (2023). Orthogonal Policy Learning Under Ambiguity

    2023

  3. [11]

    Dantzig, G. B. (1951). Maximization of a Linear Function of Variables Subject to Linear Inequalities . In Koopmans, T. C., editor, Activity Analysis of Production and Allocation , pages 339--347. Wiley & Chapman-Hall

    1951

  4. [12]

    Duarte, G., , Noam, F., , Dean, K., , Jonathan, M., , and Shpitser, I. (2024). An Automated Approach to Causal Inference in Discrete Settings . Journal of the American Statistical Association , 119(547):1778--1793

    2024

  5. [13]

    P., Allen, H., Baicker, K., and Oregon Health Study Group (2012)

    Finkelstein, A., Taubman, S., Wright, B., Bernstein, M., Gruber, J., Newhouse, J. P., Allen, H., Baicker, K., and Oregon Health Study Group (2012). The Oregon Health Insurance Experiment : Evidence from the First Year . The Quarterly Journal of Economics , 127(3):1057--1106

    2012

  6. [14]

    E., Sachs, M

    Gabriel, E. E., Sachs, M. C., and Jensen, A. K. (2024). Sharp symbolic nonparametric bounds for measures of benefit in observational and imperfect randomized studies with ordinal outcomes. Biometrika , 111(4):1429--1436

    2024

  7. [15]

    Han, S. (2021). Optimal Dynamic Treatment Regimes and Partial Welfare Ordering . Journal of the American Statistical Association

    2021

  8. [16]

    Imbens, G. W. and Manski, C. F. (2004). Confidence Intervals for Partially Identified Parameters . Econometrica , 72(6):1845--1857

    2004

  9. [17]

    Ji, W., Lei, L., and Spector, A. (2024). Model- Agnostic Covariate - Assisted Inference on Partially Identified Causal Effects . arXiv:2310.08115

    2024 arXiv

  10. [18]

    and Janson, L

    Jiang, Y. and Janson, L. (2025). Semiparametric Inference for Partially Identifiable Data Fusion Estimands via Double Machine Learning . arXiv:2502.05319

    2025 arXiv

  11. [19]

    Johnson, S. G. (2007). The NLopt nonlinear-optimization package. https://github.com/stevengj/nlopt

    2007

  12. [20]

    E., Sjölander, A., and Sachs, M

    Jonzon, G., Gabriel, E. E., Sjölander, A., and Sachs, M. C. (2025). Adding covariates to bounds: What is the question? arXiv:2502.03156 [stat]

    2025 arXiv

  13. [21]

    Kallus, N. (2022). What s the harm? sharp bounds on the fraction negatively affected by treatment. In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., and Oh, A., editors, Advances in Neural Information Processing Systems , volume 35, pages 15996--16009. Curran Ass...

    2022

  14. [22]

    and Zhou, A

    Kallus, N. and Zhou, A. (2021). Minimax-optimal policy learning under unobserved confounding . Management Science , 67(5):2870--2890

    2021

  15. [23]

    Kennedy, E. H. (2023). Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics , 17(2):3008--3049

    2023

  16. [24]

    Kennedy, E. H. (2024). Semiparametric Doubly Robust Targeted Double Machine Learning : A Review . In Handbook of Statistical Methods for Precision Medicine . Chapman and Hall/CRC

    2024

  17. [25]

    H., Balakrishnan, S., and G'Sell, M

    Kennedy, E. H., Balakrishnan, S., and G'Sell, M. (2020). Sharp instruments for classifying compliers and generalizing causal effects. The Annals of Statistics , 48(4):2008--2030

    2020

  18. [26]

    Klatt, M., Munk, A., and Zemel, Y. (2022). Limit laws for empirical optimal solutions in random linear programs. Annals of Operations Research , 315(1):251--278

    2022

  19. [27]

    Lanners, Q., Rudin, C., Volfovsky, A., and Parikh, H. (2025). Data Fusion for Partial Identification of Causal Effects . arXiv:2505.24296 [stat]

    2025 arXiv

  20. [28]

    and Talagrand, M

    Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces . Springer, Berlin, Heidelberg

    1991

  21. [29]

    W., Bonvini, M., Zeng, Z., Keele, L., and Kennedy, E

    Levis, A. W., Bonvini, M., Zeng, Z., Keele, L., and Kennedy, E. H. (2023). Covariate-assisted bounds on causal effects with instrumental variables. arXiv:2301.12106

    2023 arXiv

  22. [30]

    Liu, S., Bunea, F., and Niles-Weed, J. (2025). Beyond entropic regularization: Debiased Gaussian estimators for discrete optimal transport and general linear programs. arXiv:2505.04312 [math]

    2025 arXiv

  23. [31]

    Luedtke, A. R. and Van Der Laan , M. J. (2016). Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy . Annals of Statistics , 44(2):713--742

    2016

  24. [32]

    Mangasarian, O. L. and Shiau, T.-H. (1987). Lipschitz Continuity of Solutions of Linear Inequalities , Programs and Complementarity Problems . SIAM Journal on Control and Optimization , 25(3):583--595

    1987

  25. [33]

    Manski, C. F. (2004). Statistical Treatment Rules for Heterogeneous Populations . Econometrica , 72(4):1221--1246

    2004

  26. [34]

    Manski, C. F. (2005). Social Choice with Partial Knowledge of Treatment Response . Princeton University Press

    2005

  27. [35]

    Manski, C. F. (2007). Minimax-regret treatment choice with missing outcome data . Journal of Econometrics , 139(1):105--115

    2007

  28. [36]

    Manski, C. F. (2011). Choosing treatment policies under ambiguity . Annual Review of Economics , 3:25--49

    2011

  29. [37]

    Moulin, H. (1988). Axioms of Cooperative Decision Making . Econometric Society Monographs . Cambridge University Press, Cambridge

    1988

  30. [38]

    and Zhang, B

    Pu, H. and Zhang, B. (2021). Estimating optimal treatment rules with an instrumental variable: A partial identification learning approach . Journal of the Royal Statistical Society Series B , pages 1--28

    2021

  31. [39]

    M., Rotnitzky, A., and Ping Zhao, L

    Robins, J. M., Rotnitzky, A., and Ping Zhao, L. (1994). Estimation of Regression Coefficients When Some Regressors are not Always Observed . Journal of the American Statistical Association , 89(427):846--866

    1994

  32. [40]

    Rosenbaum, P. R. and Rubin, D. B. (1983). The Central Role of the Propensity Score in Observational Studies for Causal Effects . Biometrika , 70(1):41--55

    1983

  33. [41]

    C., , Gustav, J., , Arvid, S., , and Gabriel, E

    Sachs, M. C., , Gustav, J., , Arvid, S., , and Gabriel, E. E. (2023). A General Method for Deriving Tight Symbolic Bounds on Causal Effects . Journal of Computational and Graphical Statistics , 32(2):567--576

    2023

  34. [42]

    Semenova, V. (2024). Aggregated Intersection Bounds and Aggregated Minimax Values . arXiv:2303.00982

    2024 arXiv

  35. [43]

    Sinkhorn, R. (1967). Diagonal Equivalence to Matrices with Prescribed Row and Column Sums . The American Mathematical Monthly , 74(4):402--405

    1967

  36. [44]

    Song, K. (2014). Point decisions for interval-defined parameters. Econometric Theory , 96(2):334--356

    2014

  37. [45]

    Spielman, D. A. and Teng, S.-H. (2004). Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM , 51(3):385--463

    2004

  38. [46]

    Stoye, J. (2012). Minimax regret treatment choice with covariates or with limited validity of experiments . Journal of Econometrics , 166(1):138--156

    2012

  39. [47]

    L., Allen, H

    Taubman, S. L., Allen, H. L., Wright, B. J., Baicker, K., and Finkelstein, A. N. (2014). Medicaid Increases Emergency - Department Use : Evidence from Oregon 's Health Insurance Experiment . Science , 343(6168):263--268

    2014

  40. [48]

    Wainwright, M. J. (2019). High-Dimensional Statistics: A Non-Asymptotic Viewpoint . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press

    2019

  41. [49]

    Weed, J. (2018). An explicit analysis of the entropic penalty in linear programming. In Proceedings of the 31st Conference On Learning Theory , pages 1841--1855. PMLR. ISSN: 2640-3498

    2018

  42. [50]

    Zhang, Y., Ben-Michael, E., and Imai, K. (2023). Safe Policy Learning under Regression Discontinuity Designs with Multiple Cutoffs

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.