Policy Evaluation with Latent Confounders via Optimal Balance

Andrew Bennett; Nathan Kallus

REVIEW 3 major objections 6 minor 42 references

Policy Evaluation with Latent Confounders via Optimal Balance

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that latent confounding can be overcome in offline policy evaluation by weighting observations to adversarially balance a class of mean outcome functions that is assumed to contain the truth.

desk verdict The proxy-balancing theory is a real step forward, but the printed algorithm has index errors that disconnect it from the consistency theorem; fixable, but as written the abstract's claim is unsupported. read the letter →

arxiv 1908.01920 v1 pith:DZY5BYOC submitted 2019-08-06 stat.ML cs.LG

Andrew Bennett , Nathan Kallus This is my paper · ORCID

classification stat.MLcs.LG

keywords policyevaluationlatentconfoundersproxyvariablesimportanceweightingadversarialbalanceoff-policyreproducingkernelHilbertspaceconditionalmeansquareerror

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

Offline policy evaluation usually needs the no-unobserved-confounding assumption, which often fails in practice. This paper tries to establish that consistency is still possible when the analyst has proxies for latent confounders and an oracle for their conditional density given observables and treatment. It first proves that no single importance-weight function can be unbiased for all outcome models in this setting, then shows that an adversarially chosen weighting scheme can nonetheless drive the conditional mean square error to $O(1/n)$, yielding a root-$n$ consistent estimator of policy value. If the claim is right, data logged under one treatment rule can be re-weighted to evaluate new policies in medicine and other high-stakes settings without fitting outcome regressions.

What carries the argument

The central object is the adversarial balance objective $J(W,\mu) = \left(\frac{1}{n}\sum_{i=1}^n \sum_{t=1}^m f_{it}\nu_t(X_i,T_i)\right)^2 + \frac{2\sigma^2}{n^2}\|W\|_2^2$, with $f_{it}=W_i\delta_{T_i t}-\pi_t(X_i)$; Theorem 2 bounds the conditional MSE of the weighted estimator by $2J(W,\mu)+O(1/n)$. The paper minimizes $\sup_{\mu\in\mathcal{F}} J(W,\mu)$ rather than the unknown $J(W,\mu)$. The proof machinery is a minimax interchange, a dual calculation showing the minimum-norm weights that zero the bias term have controlled norm under Assumptions 4 to 10, and Rademacher-complexity control of the uniform deviation $\sup_{\mu\in\mathcal{F}} |E_n[\nu_T(X,T)^2]-E[\nu_T(X,T)^2]|$. For implementation, when $\mathcal{F}$ is an RKHS ball the supremum over $\mu$ becomes a quadratic form in the matrix $Q_{ij} = E[K(Z_i, Z_j')]$ with $Z_i,Z_i'$ shadow draws from the posterior density $\phi(z;x_i,t_i)$, turning the objective into a quadratic program.

What would settle it

In the paper's own simulation, replace the true posterior density $\phi(z;x,t)$ with a deliberately wrong one—for instance, omit treatment from the conditioning or shift the posterior mean—and check whether the bias of the optimal balancing weights fails to shrink as $n$ grows; that would confirm that the oracle assumption carries the consistency guarantee.

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Extended reading notes

Core claim

Under Assumptions 4 to 10, with the true mean outcome vector $\mu$ in a function class $\mathcal{F}$, the adversarially balanced estimator $\hat{\tau}^{\pi}_{W^*}$ satisfies $\hat{\tau}^{\pi}_{W^*} = \tau^{\pi} + O_p(1/\sqrt{n})$ (Theorem 3). The objective $J(W,\mu)$ upper-bounds the conditional MSE of the weighted estimator, and the key step is showing $\inf_W \sup_{\mu\in\mathcal{F}} J(W,\mu) = O(1/n)$ via a minimax exchange and uniform convergence over $\mathcal{F}$. The paper also proves an impossibility result: with latent confounding, weights that are unbiased for every outcome model would have to depend on the unknown outcome regressions $\nu_t(x,t')$, so no universal IPS-style weights exist. When $\mathcal{F}$ is the unit ball of a bounded Mercer reproducing kernel Hilbert space, the supremum over $\mathcal{F}$ becomes a closed quadratic form and the weights come from a quadratic program.

Load-bearing premise

The guarantee collapses if the oracle for the conditional density $\phi(z;x,t)$ of the latent confounders given observed covariates and treatment is misspecified, because the balancing objective then no longer controls the true mean outcome functions and bias need not vanish.

Editorial extensions

If this is right

If the class $\mathcal{F}$ contains the true outcome regression, weighted policy evaluation is consistent even when unconfoundedness given $X$ fails.
No fixed weighting recipe can be unbiased for all outcome models once confounders are latent; unbiasedness would require knowing the outcome regressions, so adversarial balance is the workable alternative.
With a bounded Mercer kernel such as the Gaussian kernel, the adversarial objective becomes a tractable quadratic program, making the method usable on real datasets.
In the paper's simulations, benchmarks that assume ignorability given $X$ show nonzero asymptotic bias, while bias of the balancing weights shrinks with sample size.
The same weights plug into doubly robust estimators, and consistency of those estimators follows whenever the weighting side is consistent.

Reading between the lines

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

The oracle assumption is doing real work: if $\phi(z;x,t)$ is estimated from the same logged data, Theorem 3 does not by itself cover estimation error, so the practical guarantee depends on identifiability of the latent model.
A natural extension is to feed the same adversarial objective with posterior samples from more flexible fitted models and quantify how error in the posterior propagates into $Q$ and into the final estimate.
The impossibility result suggests that methods using only observed covariates $X$ cannot achieve universal unbiasedness under this causal graph; some proxy-model assumption is unavoidable.
The same minimax objective could be instantiated with other function classes, such as neural networks, by replacing the closed-form RKHS supremum with an inner-loop adversarial fit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: substantive theory, real contribution, but the algorithm as printed does not implement the theorem. In Theorem 5 and Algorithm 1, the linear coefficient a_i is defined with swapped indices, and step 4 of Algorithm 1 estimates Q_ij using samples only from unit i instead of from both i and j. Both are clearly typos—the correct forms are in the proof in A.7—but as printed, the QP being solved is not the adversarial objective analyzed in A.7, so Theorem 3's consistency guarantee does not attach to the algorithm as written. This is a fixable presentation error, not a flaw in the underlying theory.

What is actually new: Theorem 1 is a clean impossibility result—no single weight function can be unbiased for all outcome models when only proxies are available. The proposed remedy, adversarial balancing over the latent posterior, genuinely extends Kallus (2018a) from the unconfounded setting to latent confounders, and the proof that this achieves root-n consistency under RKHS classes (Theorems 3–5, plus the empirical-process arguments in A.5) is a real technical achievement. The appendix is thorough, the main proofs check out, and the experiments are an honest proof-of-concept: they assume the posterior oracle and show the method beating ignorability-based baselines.

Soft spots, in proportion: the index bugs are the load-bearing issue because the paper's central claim is that Algorithm 1 'provably guarantees consistency.' As printed, that claim is unsupported. The fix is straightforward, but the experiments should also say which version of the algorithm was actually run. The other soft spot is the oracle assumption for the latent posterior φ(z;x,t) and the requirement that μ be in F. The paper is upfront that it does not address identification of the latent model, so this is a stated boundary rather than a hidden flaw—but it does limit the practical reach.

Who this is for: anyone working on offline policy evaluation, causal inference with proxies, or balancing weights. The theory deserves a serious referee; the bugs are exactly what referees are for. Send it to peer review, with instructions that the algorithm/theorem mismatch must be corrected.

Referee Report

3 major / 6 minor

Summary. The paper studies off-policy evaluation of contextual bandit policies from logged data in the presence of latent confounders. The authors assume that the confounders Z render potential outcomes independent of (X,T), that X is a proxy for Z, and that a correct model of the posterior density φ(z;x,t) is available as an oracle. They first prove that no single importance weight can be unbiased for all outcome models (Theorem 1), then propose to select weights by minimizing the worst-case conditional mean squared error over a class F of candidate outcome regressions. Under regularity and complexity assumptions on F and assuming the true regression lies in F, they prove the resulting estimator is root-n consistent (Theorems 2 and 3). For RKHS classes they show the adversarial objective reduces to a quadratic program (Theorem 5) and provide Algorithm 1, estimating the Gram matrix by posterior sampling. Synthetic experiments compare the method with IPS, ignorability-based balancing, and direct regression methods.

Significance. This is a relevant and potentially useful contribution. The impossibility result (Theorem 1) clarifies why proxy-based balancing differs from standard IPS, and the adversarial-balance formulation, together with the consistency theorem (Theorem 3), offers a principled way to avoid fitting outcome regressions when confounders are latent. The appendix contains detailed proofs, and the experiments, while on a single synthetic scenario, show the expected qualitative behavior. However, the printed algorithm does not implement the theorem's objective: the Q-matrix estimator in Algorithm 1 uses the wrong posterior samples, and the linear coefficient in Theorem 5 and Algorithm 1 has a load-bearing index swap. In addition, a bound in the proof of Theorem 2 is not valid as stated. These issues are fixable but are central to the claim that Algorithm 1 'provably guarantees consistency,' so the paper needs a major revision.

major comments (3)

[Section 4.2, Algorithm 1, line 4] The printed formula computes Q_ij = (1/B^2) ∑_{b=1}^B ∑_{c=1}^B K(Z_i^b, Z_i^c), using posterior samples from unit i in both arguments. This estimates E[K(Z_i, Z_i')], not the off-diagonal kernel expectation Q_ij = E[K(Z_i, Z_j')] defined in Theorem 5, where Z_j' is drawn from the posterior of unit j. The correct estimator, matching the proof in Appendix A.7, is (1/B^2) ∑_{b,c} K(Z_i^b, Z_j^c). As printed, Algorithm 1 does not minimize the adversarial objective, so the consistency guarantee of Theorem 3 does not apply to the weights it produces.
[Section 4.2, Theorem 5 and Algorithm 1, line 5] The linear term is printed as a_i = (2/n^2) ∑_{j=1}^n Q_ij π_{T_j}(X_i). Expanding the objective in Appendix A.7 gives the coefficient of W_j as (2/n^2) ∑_{i=1}^n Q_ij π_{T_j}(X_i); using symmetry of Q, this is equivalent to a_i = (2/n^2) ∑_{j=1}^n Q_ij π_{T_i}(X_j). The printed version evaluates the policy of unit j at the covariate of unit i, whereas the correct term evaluates the policy of unit i at the covariate of unit j. Consequently the QP solved by Algorithm 1 is not the supremum over F_K of J(W, μ), and the printed claim that Algorithm 1 provably guarantees consistency is unsupported.
[Appendix A.2, proof of Theorem 2] The step bounding V[∑_t f_it μ_t(Z_i) | X,T] by max_t f_it^2 V[μ_t(Z_i) | X,T] does not follow from the lemma stated immediately above, which requires coefficients that sum to one; here ∑_t f_it = W_i - 1. The claimed inequality is false in general (for example, two negatively correlated variables with equal-magnitude coefficients give variance exceeding max_t f_it^2). A correct argument introduces sums of squares of f_it or (W_i - 1)^2 terms, changing the constants in J. The O(1/n) conclusion is likely still recoverable, but the proof as written is not valid.

minor comments (6)

[Section 2.1, Assumption 1] The word 'independant' should be 'independent'.
[Section 4.2, Algorithm 1] The algorithm omits the global 1/n^2 factors in G and a_i that appear in Theorem 5; this is equivalent to multiplying the whole objective by n^2 and leaves the minimizer unchanged, but the convention should be stated so that the hyperparameter Γ is interpreted consistently.
[Appendix A.7] In the derivation of sup_{μ_t ∈ F^K_t}, the display containing ⟨T_K s, T_K s⟩_K^2 / ||T_K s||_K should instead be ||T_K s||_K^2; the following line is correct, so this appears to be a typographical slip.
[Section 2.3] The sentence 'Some recent work on this problem involves using techniques such as matrix completion (Kallus, 2018a)' cites Kallus (2018a), which is the balanced policy evaluation paper; the relevant matrix-factorization reference appears to be Kallus, Mao, and Udell (2018).
[Section 3.3] The claim that Assumptions 4 to 10 are satisfied by 'families of neural networks' is stated without proof or reference; only the RKHS case is established in Section 4.
[Section 5.2, Table 1 caption] The caption reads 'Convergence of RMSE for for policy evaluation'; the duplicated 'for' should be removed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the adversarial balancing estimator is not fitted to outcomes, and the consistency theorem rests on the posterior-oracle and regularity assumptions rather than on a fitted input.

full rationale

The weighted estimator in Algorithm 1 is constructed from observed covariates, treatments, the posterior oracle phi, the kernel K, and the policy pi; it never uses the outcome values Y. The consistency argument is a genuine minimax bound: Theorem 2 bounds the conditional MSE by 2J(W,mu)+O_p(1/n); Theorem 5 identifies sup over mu in F_K of J(W,mu) as a quadratic program in W; and Lemma 4 proves inf_W sup over mu in F of J(W,mu) = O_p(1/n) under Assumptions 4 to 10. None of these identities defines the estimand in terms of the estimator or fits a parameter to the quantity being predicted. The condition mu in F is a coverage assumption rather than a fitted value, and Theorem 1 independently shows that no single weight can be unbiased for all outcome functions, which provides non-circular motivation for the adversarial formulation. The only self-citations used as logical support are Kallus (2016, Lemma 31) for the elementary implication from conditional mean squared error O_p(1/n) to root-n consistency, and Kallus (2018a) as related-work inspiration; the former is a standard one-line probabilistic fact whose assumptions do not include the target result, so it is not load-bearing. A separate correctness concern is that Algorithm 1 and Theorem 5 as printed write a_i = 2/n^2 times sum over j of Q_ij pi_{T_j}(X_i), whereas the expansion in Appendix A.7 yields the linear coefficient for W_i as 2/n^2 times sum over j of Q_ji pi_{T_i}(X_j), which by symmetry of Q equals 2/n^2 times sum over j of Q_ij pi_{T_i}(X_j); the subscripts on the policy term are swapped, and Algorithm 1 also omits the 1/n^2 normalization. This is a technical bug in the printed algorithm-to-theorem match, not a circularity: the proof of Theorem 5 is self-contained and the fix is explicit, so it does not raise the circularity score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or conceptual entities. The latent Z and its posterior φ are part of the problem setting and the assumed oracle, respectively. The free parameters are the kernel, the RKHS norm bound Γ, and the Monte Carlo sample size B in the algorithm.

free parameters (3)

Γ (regularization matrix) = γ ∈ {0.001, 0.2, 1.0, 5.0} in experiments (no selection rule)
Controls the RKHS norm bound in the kernel class; if set too small, the true μ may fall outside F and consistency is not guaranteed. Treated as a hyperparameter in the experiments.
Kernel K = unspecified (assumed Gaussian)
The kernel defines the RKHS class F_K; the paper only requires K to be bounded and Mercer (Theorem 4) and notes the Gaussian kernel satisfies this, but the bandwidth is not specified.
Posterior sample size B = B=50
Used in Algorithm 1 to approximate Q_{ij}; larger B reduces Monte Carlo error in the QP inputs but is not part of the theory.

assumptions (6)

domain assumption Assumption 1: Y(t) ⊥ (X,T) | Z (Z are true confounders)
This is the key identification condition; it states that all confounding is captured by Z, and X is only a proxy for Z.
standard math Assumption 2: E[e_t^{-2}(Z)] < ∞ (weak overlap)
Ensures inverse propensity terms are manageable; standard in weighting methods.
standard math Assumption 3: Var(Y(t)|X,T) ≤ σ² (bounded variance)
Used in Theorem 2 to bound the variance term.
domain assumption Assumptions 4-10 on the function class F (normed, star-shaped, convex compact, square-integrable, nondegenerate, bounded, low Rademacher complexity)
Regularity conditions that make the adversarial balance objective tractable; satisfied by RKHS balls (Theorem 4).
domain assumption Oracle for posterior density φ(z;x,t) and identification of the latent variable model
The paper assumes we can evaluate φ exactly (Section 2.1). Identification of the latent model from proxies is delegated to prior work (Miao et al. 2016, etc.). This is the main practical assumption.
domain assumption μ ∈ F (the true outcome regression lies in the function class)
Needed for Theorem 3; if μ is outside F, the worst-case bound does not cover it. This is standard in nonparametric balancing and is controlled by the choice of Γ.

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Cite this review

Pith. "Pith review of Policy Evaluation with Latent Confounders via Optimal Balance." pith.science (2026). https://pith.science/paper/DZY5BYOC

@misc{pith2026190801920,
  author       = {Pith},
  title        = {Pith review of: Policy Evaluation with Latent Confounders via Optimal Balance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZY5BYOC}},
  note         = {Machine review of arXiv:1908.01920}
}

read the original abstract

Evaluating novel contextual bandit policies using logged data is crucial in applications where exploration is costly, such as medicine. But it usually relies on the assumption of no unobserved confounders, which is bound to fail in practice. We study the question of policy evaluation when we instead have proxies for the latent confounders and develop an importance weighting method that avoids fitting a latent outcome regression model. We show that unlike the unconfounded case no single set of weights can give unbiased evaluation for all outcome models, yet we propose a new algorithm that can still provably guarantee consistency by instead minimizing an adversarial balance objective. We further develop tractable algorithms for optimizing this objective and demonstrate empirically the power of our method when confounders are latent.

Figures

Figures reproduced from arXiv: 1908.01920 by the authors.

**Figure 1.** DAG representation of problem. We encapsulate the assumption that Z are sufficient for unconfoundedness and that X is a proxy for Z in the following assumption [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 36 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...
[2]

W., and Wager, S

Athey, S., Imbens, G. W., and Wager, S. Approximate residual balancing: debiased inference of average treatment effects in high dimensions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80 0 (4): 0 597--623, 2018

work page 2018
[3]

Austin, P. C. and Stuart, E. A. Moving towards best practice when using inverse probability of treatment weighting (iptw) using the propensity score to estimate causal treatment effects in observational studies. Statistics in medicine, 34 0 (28): 0 3661--3679, 2015

work page 2015
[4]

M., and Zhuo, Y

Bertsimas, D., Kallus, N., Weinstein, A. M., and Zhuo, Y. D. Personalized diabetes management using electronic medical records. Diabetes care, 40 0 (2): 0 210--217, 2017

work page 2017
[5]

and Langford, J

Beygelzimer, A. and Langford, J. The offset tree for learning with partial labels. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp.\ 129--138. ACM, 2009

work page 2009
[6]

Q., Charles, D

Bottou, L., Peters, J., Candela, J. Q., Charles, D. X., Chickering, M., Portugaly, E., Ray, D., Simard, P. Y., and Snelson, E. Counterfactual reasoning and learning systems: the example of computational advertising. Journal of Machine Learning Research, 14 0 (1): 0 3207--3260, 2013

work page 2013
[7]

and Kuroki, M

Cai, Z. and Kuroki, M. On identifying total effects in the presence of latent variables and selection bias. In Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence, pp.\ 62--69. AUAI Press, 2008

work page 2008
[8]

D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A

Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A. Stan: A probabilistic programming language. Journal of statistical software, 76 0 (1), 2017

work page 2017

Show all 42 references

[9]

Double machine learning for treatment and causal parameters

Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., and Hansen, C. Double machine learning for treatment and causal parameters. arXiv preprint arXiv:1608.00060, 2016

2016 arXiv
[10]

Doubly robust policy evaluation and learning

Dud \' k, M., Langford, J., and Li, L. Doubly robust policy evaluation and learning. In Proceedings of the 28th International Conference on International Conference on Machine Learning, pp.\ 1097--1104. Omnipress, 2011

2011
[11]

K., Cole, S

Edwards, J. K., Cole, S. R., and Westreich, D. All your data are always missing: incorporating bias due to measurement error into the potential outcomes framework. International journal of epidemiology, 44 0 (4): 0 1452--1459, 2015

2015
[12]

Elliott, M. R. Model averaging methods for weight trimming. Journal of official statistics, 24 0 (4): 0 517, 2008

2008
[13]

Frost, P. A. Proxy variables and specification bias. The review of economics and Statistics, pp.\ 323--325, 1979

1979
[14]

Ionides, E. L. Truncated importance sampling. Journal of Computational and Graphical Statistics, 17 0 (2): 0 295--311, 2008

2008
[15]

Generalized optimal matching methods for causal inference

Kallus, N. Generalized optimal matching methods for causal inference. arXiv preprint arXiv:1612.08321, 2016

2016 arXiv
[16]

Recursive partitioning for personalization using observational data

Kallus, N. Recursive partitioning for personalization using observational data. In International Conference on Machine Learning (ICML), pp.\ 1789--1798, 2017 a

2017
[17]

A framework for optimal matching for causal inference

Kallus, N. A framework for optimal matching for causal inference. In Artificial Intelligence and Statistics (AISTATS), pp.\ 372--381, 2017 b

2017
[18]

Balanced policy evaluation and learning

Kallus, N. Balanced policy evaluation and learning. In Advances in Neural Information Processing Systems, pp.\ 8895--8906, 2018 a

2018
[19]

Optimal a priori balance in the design of controlled experiments

Kallus, N. Optimal a priori balance in the design of controlled experiments. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80 0 (1): 0 85--112, 2018 b

2018
[20]

and Zhou, A

Kallus, N. and Zhou, A. Policy evaluation and optimization with continuous treatments. In International Conference on Artificial Intelligence and Statistics, pp.\ 1243--1251, 2018

2018
[21]

Causal inference with noisy and missing covariates via matrix factorization

Kallus, N., Mao, X., and Udell, M. Causal inference with noisy and missing covariates via matrix factorization. In Advances in Neural Information Processing Systems, pp.\ 6921--6932, 2018

2018
[22]

Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

2014 arXiv
[23]

Kube, A., Das, S., and Fowler, P. J. Allocating interventions based on predicted outcomes: A case study on homelessness services. In Proceedings of the AAAI Conference on Artificial Intelligence, 2019

2019
[24]

and Pearl, J

Kuroki, M. and Pearl, J. Measurement bias and effect restoration in causal inference. Biometrika, 101 0 (2): 0 423--437, 2014

2014
[25]

and Talagrand, M

Ledoux, M. and Talagrand, M. Probability in Banach Spaces: isoperimetry and processes. Springer Science & Business Media, 2013

2013
[26]

Unbiased offline evaluation of contextual-bandit-based news article recommendation algorithms

Li, L., Chu, W., Langford, J., and Wang, X. Unbiased offline evaluation of contextual-bandit-based news article recommendation algorithms. In Proceedings of the fourth ACM international conference on Web search and data mining, pp.\ 297--306. ACM, 2011

2011
[27]

M., Sontag, D., Zemel, R., and Welling, M

Louizos, C., Shalit, U., Mooij, J. M., Sontag, D., Zemel, R., and Welling, M. Causal effect inference with deep latent-variable models. In Advances in Neural Information Processing Systems, pp.\ 6446--6456, 2017

2017
[28]

Lunceford, J. K. and Davidian, M. Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. Statistics in medicine, 23 0 (19): 0 2937--2960, 2004

2004
[29]

Offline policy evaluation across representations with applications to educational games

Mandel, T., Liu, Y.-E., Levine, S., Brunskill, E., and Popovic, Z. Offline policy evaluation across representations with applications to educational games. In Proceedings of the International Conference on Autonomous Agents and Multi-agent Systems, pp.\ 1077--1084. Internation...

2014
[30]

On the performance of kernel classes

Mendelson, S. On the performance of kernel classes. Journal of Machine Learning Research, 4 0 (Oct): 0 759--771, 2003

2003
[31]

Miao, W., Geng, Z., and Tchetgen, E. T. Identifying causal effects with proxy variables of an unmeasured confounder. arXiv preprint arXiv:1609.08816, 2016

2016 arXiv
[32]

Causality: models, reasoning and inference

Pearl, J. Causality: models, reasoning and inference. Cambridge University Press, 2000

2000
[33]

On measurement bias in causal inference

Pearl, J. On measurement bias in causal inference. arXiv preprint arXiv:1203.3504, 2012

2012 arXiv
[34]

and Murphy, S

Qian, M. and Murphy, S. A. Performance guarantees for individualized treatment rules. Annals of statistics, 39 0 (2): 0 1180, 2011

2011
[35]

Robins, J. M. Robust estimation in sequentially ignorable missing data and causal inference models. In Proceedings of the American Statistical Association, pp.\ 6--10, 1999

1999
[36]

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

Robins, J. M., Rotnitzky, A., and Zhao, L. P. Estimation of regression coefficients when some regressors are not always observed. Journal of the American statistical Association, 89 0 (427): 0 846--866, 1994

1994
[37]

O., Rotnitzky, A., and Robins, J

Scharfstein, D. O., Rotnitzky, A., and Robins, J. M. Adjusting for nonignorable drop-out using semiparametric nonresponse models. Journal of the American Statistical Association, 94 0 (448): 0 1096--1120, 1999

1999
[38]

and Joachims, T

Swaminathan, A. and Joachims, T. Counterfactual risk minimization: Learning from logged bandit feedback. In ICML, pp.\ 814--823, 2015 a

2015
[39]

and Joachims, T

Swaminathan, A. and Joachims, T. The self-normalized estimator for counterfactual learning. In Advances in Neural Information Processing Systems, pp.\ 3231--3239, 2015 b

2015
[40]

Wickens, M. R. A note on the use of proxy variables. Econometrica: Journal of the Econometric Society, pp.\ 759--761, 1972

1972
[41]

Wooldridge, J. M. On estimating firm-level production functions using proxy variables to control for unobservables. Economics Letters, 104 0 (3): 0 112--114, 2009

2009
[42]

The covering number in learning theory

Zhou, D.-X. The covering number in learning theory. Journal of Complexity, 18 0 (3): 0 739--767, 2002

2002

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

[2] [2]

W., and Wager, S

Athey, S., Imbens, G. W., and Wager, S. Approximate residual balancing: debiased inference of average treatment effects in high dimensions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80 0 (4): 0 597--623, 2018

work page 2018

[3] [3]

Austin, P. C. and Stuart, E. A. Moving towards best practice when using inverse probability of treatment weighting (iptw) using the propensity score to estimate causal treatment effects in observational studies. Statistics in medicine, 34 0 (28): 0 3661--3679, 2015

work page 2015

[4] [4]

M., and Zhuo, Y

Bertsimas, D., Kallus, N., Weinstein, A. M., and Zhuo, Y. D. Personalized diabetes management using electronic medical records. Diabetes care, 40 0 (2): 0 210--217, 2017

work page 2017

[5] [5]

and Langford, J

Beygelzimer, A. and Langford, J. The offset tree for learning with partial labels. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp.\ 129--138. ACM, 2009

work page 2009

[6] [6]

Q., Charles, D

Bottou, L., Peters, J., Candela, J. Q., Charles, D. X., Chickering, M., Portugaly, E., Ray, D., Simard, P. Y., and Snelson, E. Counterfactual reasoning and learning systems: the example of computational advertising. Journal of Machine Learning Research, 14 0 (1): 0 3207--3260, 2013

work page 2013

[7] [7]

and Kuroki, M

Cai, Z. and Kuroki, M. On identifying total effects in the presence of latent variables and selection bias. In Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence, pp.\ 62--69. AUAI Press, 2008

work page 2008

[8] [8]

D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A

Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A. Stan: A probabilistic programming language. Journal of statistical software, 76 0 (1), 2017

work page 2017