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James M. Robins

Publications and source records attributed to James M. Robins.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Value Inference for Reinforcement Learning

We study offline inference for the optimal value in reinforcement learning. Two new nuisances are derived as fixed points of a self-induced Bellman equation, in which we approximate the maximum Bellman operator by its softmax correspondence. We propose a debiased estimator through the Neyman orthogonality and establish its asymptotic normality under diverging horizons even when the behavior policy changes with time, as long as the nuisances have the statistical rates that can be achieved by many machine learning methods. We provide a concrete estimating procedure for these nuisances and show they can lead to valid inference. Synthetic experiments validate the numerical performance of our inference method, and we implement it in real-life decision-making problems, including bike repositioning and AI agentic tool use.

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arXiv2026

Improved Variance Estimation in Homoskedastic Nonparametric Random-Design Regression via a Two-Scale Approach

We study estimation of a constant conditional variance $\sigma^2$ in nonparametric regression with a $d$-dimensional random design. This is an important problem, and similar questions arise in causal inference. The regression function is $\beta_b$-H\"older smooth, the design density is $\beta_g$-H\"older smooth and bounded above and away from zero, and we consider the nonparametric regime $\beta_b>1$ and $d>4\beta_b$. Set $\beta_g^\star=\beta_b(1-4\beta_b/d)/\{1+2\beta_b/d+8(\beta_b/d)^2\}$. We give an estimator whose mean squared error is upper bounded by $Cn^{-4(\beta_b+1)/(d+4)}$ in the low-regularity regime when $0<\beta_g\leq\beta_g^\star$. The low-regularity branch is based on a new two-scale construction: the covariate space is partitioned into cells, the local polynomial trend is projected out within each suitable cell, and the squared normalized contrast from one eligible close pair per cell is averaged across cells. In the high regularity regime when $\beta_g>\beta_g^\star$, a higher-order influence function estimator of Robins, Li, Tchetgen Tchetgen, and van der Vaart (2008) provides the rate $Cn^{-8\beta_b/(d+4\beta_b)}$. We also give an all-pairs ridge extension, which achieves the same two-scale rate, and evaluate the methods alongside a range of existing estimators in simulations.

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arXiv2025

A note on the relation between one-step, outcome regression and IPW-type estimators of parameters with the mixed bias property

Bruns-Smith et al. (2025) established an algebraic identity between the one-step estimator and a specific outcome regression-type estimator for a class of parameters that forms a strict subset of the class introduced in Chernozhukov et al. (2022), assuming both nuisance functions are estimated as linear combinations of given features. They conjectured that this identity extends to the broader mixed bias class introduced in Rotnitzky et al. (2021). In this note, we prove their conjecture and further extend the result to allow one of the nuisance estimators to be non-linear. We also relate these findings to the work of Robins et al. (2007), who established other identities linking one-step estimators to outcome regression-type and IPW-type estimators.

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arXiv2025

A Multiplicative Instrumental Variable Model for Data Missing Not-at-Random

Instrumental variable (IV) methods offer a valuable approach to account for outcome data missing not-at-random. A valid missing data instrument is a measured factor which (i) predicts the nonresponse process and (ii) is independent of the outcome in the underlying population. For point identification, all existing IV methods for missing data including the celebrated Heckman selection model, a priori restrict the extent of selection bias on the outcome scale, therefore potentially understating uncertainty due to missing data. In this work, we introduce an IV framework which allows the degree of selection bias on the outcome scale to remain completely unrestricted. The new approach instead relies for identification on (iii) a key multiplicative selection model, which posits that the instrument and any hidden common correlate of selection and the outcome, do not interact on the multiplicative scale. Interestingly, we establish that any regular statistical functional of the missing outcome is nonparametrically identified under (i)-(iii) via a single-arm Wald ratio estimand reminiscent of the standard Wald ratio estimand in causal inference. For estimation and inference, we characterize the influence function for any functional defined on a nonparametric model for the observed data, which we leverage to develop semiparametric multiply robust IV estimators. Several extensions of the methods are also considered, including the important practical setting of polytomous and continuous instruments. Simulation studies illustrate the favorable finite sample performance of proposed methods, which we further showcase in an HIV study nested within a household health survey study we conducted in Mochudi, Botswana, in which interviewer characteristics are used as instruments to correct for selection bias due to dependent nonresponse in the HIV component of the survey study.

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arXiv2025

Inference on Nonlinear Counterfactual Functionals under a Multiplicative IV Model

Instrumental variable (IV) methods play a central role in causal inference, particularly in settings where treatment assignment is confounded by unobserved variables. IV methods have been extensively developed in recent years and applied across diverse domains, from economics to epidemiology. In this work, we study the recently introduced multiplicative IV (MIV) model and demonstrate its utility for causal inference beyond the average treatment effect. In particular, we show that it enables identification and inference for a broad class of counterfactual functionals characterized by moment equations. This includes, for example, inference on quantile treatment effects. We develop methods for efficient and multiply robust estimation of such functionals, and provide inference procedures with asymptotic validity. Experimental results demonstrate that the proposed procedure performs well even with moderate sample sizes.

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arXiv2025

On the asymptotic validity of confidence sets for linear functionals of solutions to integral equations

This paper examines the construction of confidence sets for parameters defined as linear functionals of a function of W and X whose conditional mean given Z and X equals the conditional mean of another variable Y given Z and X. Many estimands of interest in causal inference can be expressed in this form, including the average treatment effect in proximal causal inference and treatment effect contrasts in instrumental variable models. We derive a necessary condition for a confidence set to be uniformly valid over a model that allows for the dependence between W and Z given X to be arbitrarily weak. Specifically, we show that for any such confidence set, there must exist some laws in the model under which, with high probability, the confidence set has a diameter greater than or equal to the diameter of the parameter's range. In particular, consistent with the weak instruments literature, Wald confidence intervals are not uniformly valid over the aforementioned model when the parameter's range is infinite. Furthermore, we argue that inverting the score test, a successful approach in that literature, generally fails for the broader class of parameters considered here. We present a method for constructing uniformly valid confidence sets in the special case where all variables, but possibly Y, are binary and discuss its limitations. Finally, we emphasize that developing uniformly valid confidence sets for the class of parameters considered in this paper remains an open problem.

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arXiv2025

The Multiplicative Instrumental Variable Model

The instrumental variable (IV) design is a common approach to address hidden confounding bias. For validity, an IV must impact the outcome only through its association with the treatment. In addition, IV identification has required a homogeneity condition such as monotonicity or no unmeasured common effect modifier between the additive effect of the treatment on the outcome, and that of the IV on the treatment. In this work, we introduce the Multiplicative Instrumental Variable Model (MIV), which encodes a condition of no multiplicative interaction between the instrument and an unmeasured confounder in the treatment propensity score model. Thus, the MIV provides a novel formalization of the core IV independence condition interpreted as independent mechanisms of action, by which the instrument and hidden confounders influence treatment uptake, respectively. As we formally establish, MIV provides nonparametric identification of the population average treatment effect on the treated (ATT) via a single-arm version of the classical Wald ratio IV estimand, for which we propose a novel class of estimators that are multiply robust and semiparametric efficient. Finally, we illustrate the methods in extended simulations and an application on the causal impact of a job training program on subsequent earnings.

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arXiv2025

Debiased Ill-Posed Regression

In various statistical settings, the goal is to estimate a function which is restricted by the statistical model only through a conditional moment restriction. Prominent examples include the nonparametric instrumental variable framework for estimating the structural function of the outcome variable, and the proximal causal inference framework for estimating the bridge functions. A common strategy in the literature is to find the minimizer of the projected mean squared error. However, this approach can be sensitive to misspecification or slow convergence rate of the estimators of the involved nuisance components. In this work, we propose a debiased estimation strategy based on the influence function of a modification of the projected error and demonstrate its finite-sample convergence rate. Our proposed estimator possesses a second-order bias with respect to the involved nuisance functions and a desirable robustness property with respect to the misspecification of one of the nuisance functions. The proposed estimator involves a hyper-parameter, for which the optimal value depends on potentially unknown features of the underlying data-generating process. Hence, we further propose a hyper-parameter selection approach based on cross-validation and derive an error bound for the resulting estimator. This analysis highlights the potential rate loss due to hyper-parameter selection and underscore the importance and advantages of incorporating debiasing in this setting. We also study the application of our approach to the estimation of regular parameters in a specific parameter class, which are linear functionals of the solutions to the conditional moment restrictions and provide sufficient conditions for achieving root-n consistency using our debiased estimator.

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arXiv2025

Using negative controls to identify causal effects with invalid instrumental variables

Many proposals for the identification of causal effects require an instrumental variable that satisfies strong, untestable unconfoundedness and exclusion restriction assumptions. In this paper, we show how one can potentially identify causal effects under violations of these assumptions by harnessing a negative control population or outcome. This strategy allows one to leverage sub-populations for whom the exposure is degenerate, and requires that the instrument-outcome association satisfies a certain parallel trend condition. We develop the semiparametric efficiency theory for a general instrumental variable model, and obtain a multiply robust, locally efficient estimator of the average treatment effect in the treated. The utility of the estimators is demonstrated in simulation studies and an analysis of the Life Span Study.

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arXiv2024

Causal and Counterfactual Views of Missing Data Models

It is often said that the fundamental problem of causal inference is a missing data problem -- the comparison of responses to two hypothetical treatment assignments is made difficult because for every experimental unit only one potential response is observed. In this paper, we consider the implications of the converse view: that missing data problems are a form of causal inference. We make explicit how the missing data problem of recovering the complete data law from the observed law can be viewed as identification of a joint distribution over counterfactual variables corresponding to values had we (possibly contrary to fact) been able to observe them. Drawing analogies with causal inference, we show how identification assumptions in missing data can be encoded in terms of graphical models defined over counterfactual and observed variables. We review recent results in missing data identification from this viewpoint. In doing so, we note interesting similarities and differences between missing data and causal identification theories.

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arXiv2024

Assumptions and Bounds in the Instrumental Variable Model

In this note we give proofs for results relating to the Instrumental Variable (IV) model with binary response $Y$ and binary treatment $X$, but with an instrument $Z$ with $K$ states. These results were originally stated in Richardson & Robins (2014), "ACE Bounds; SEMS with Equilibrium Conditions," arXiv:1410.0470.

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arXiv2024

Grace periods in comparative effectiveness studies of sustained treatments

Researchers are often interested in estimating the effect of sustained use of a treatment on a health outcome. However, adherence to strict treatment protocols can be challenging for individuals in practice and, when non-adherence is expected, estimates of the effect of sustained use may not be useful for decision making. As an alternative, more relaxed treatment protocols which allow for periods of time off treatment (i.e. grace periods) have been considered in pragmatic randomized trials and observational studies. In this article, we consider the interpretation, identification, and estimation of treatment strategies which include grace periods. We contrast natural grace period strategies which allow individuals the flexibility to take treatment as they would naturally do, with stochastic grace period strategies in which the investigator specifies the distribution of treatment utilization. We estimate the effect of initiation of a thiazide diuretic or an angiotensin-converting enzyme inhibitor in hypertensive individuals under various strategies which include grace periods.

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arXiv2023

Minimax rates for heterogeneous causal effect estimation

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference. Many methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a Holder-smooth nonparametric model, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. Our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid.

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arXiv2023

Nested Markov Properties for Acyclic Directed Mixed Graphs

Conditional independence models associated with directed acyclic graphs (DAGs) may be characterized in at least three different ways: via a factorization, the global Markov property (given by the d-separation criterion), and the local Markov property. Marginals of DAG models also imply equality constraints that are not conditional independences; the well-known ``Verma constraint'' is an example. Constraints of this type are used for testing edges, and in a computationally efficient marginalization scheme via variable elimination. We show that equality constraints like the ``Verma constraint'' can be viewed as conditional independences in kernel objects obtained from joint distributions via a fixing operation that generalizes conditioning and marginalization. We use these constraints to define, via ordered local and global Markov properties, and a factorization, a graphical model associated with acyclic directed mixed graphs (ADMGs). We prove that marginal distributions of DAG models lie in this model, and that a set of these constraints given by Tian provides an alternative definition of the model. Finally, we show that the fixing operation used to define the model leads to a particularly simple characterization of identifiable causal effects in hidden variable causal DAG models.

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arXiv2023

Potential Outcome and Decision Theoretic Foundations for Statistical Causality

In a recent paper published in the Journal of Causal Inference, Philip Dawid has described a graphical causal model based on decision diagrams. This article describes how single-world intervention graphs (SWIGs) relate to these diagrams. In this way, a correspondence is established between Dawid's approach and those based on potential outcomes such as Robins' Finest Fully Randomized Causally Interpreted Structured Tree Graphs. In more detail, a reformulation of Dawid's theory is given that is essentially equivalent to his proposal and isomorphic to SWIGs.

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arXiv2023

Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators

The class of doubly-robust (DR) functionals studied by Rotnitzky et al. (2021) is of central importance in economics and biostatistics. It strictly includes both (i) the class of mean-square continuous functionals that can be written as an expectation of an affine functional of a conditional expectation studied by Chernozhukov et al. (2022b) and (ii) the class of functionals studied by Robins et al. (2008). The present state-of-the-art estimators for DR functionals $ψ$ are double-machine-learning (DML) estimators (Chernozhukov et al., 2018). A DML estimator $\widehatψ_{1}$ of $ψ$ depends on estimates $\widehat{p} (x)$ and $\widehat{b} (x)$ of a pair of nuisance functions $p(x)$ and $b(x)$, and is said to satisfy "rate double-robustness" if the Cauchy--Schwarz upper bound of its bias is $o (n^{- 1/2})$. Were it achievable, our scientific goal would have been to construct valid, assumption-lean (i.e. no complexity-reducing assumptions on $b$ or $p$) tests of the validity of a nominal $(1 - α)$ Wald confidence interval (CI) centered at $\widehatψ_{1}$. But this would require a test of the bias to be $o (n^{-1/2})$, which can be shown not to exist. We therefore adopt the less ambitious goal of falsifying, when possible, an analyst's justification for her claim that the reported $(1 - α)$ Wald CI is valid. In many instances, an analyst justifies her claim by imposing complexity-reducing assumptions on $b$ and $p$ to ensure "rate double-robustness". Here we exhibit valid, assumption-lean tests of $H_{0}$: "rate double-robustness holds", with non-trivial power against certain alternatives. If $H_{0}$ is rejected, we will have falsified her justification. However, no assumption-lean test of $H_{0}$, including ours, can be a consistent test. Thus, the failure of our test to reject is not meaningful evidence in favor of $H_{0}$.

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arXiv2022

Deep Learning Methods for Proximal Inference via Maximum Moment Restriction

The No Unmeasured Confounding Assumption is widely used to identify causal effects in observational studies. Recent work on proximal inference has provided alternative identification results that succeed even in the presence of unobserved confounders, provided that one has measured a sufficiently rich set of proxy variables, satisfying specific structural conditions. However, proximal inference requires solving an ill-posed integral equation. Previous approaches have used a variety of machine learning techniques to estimate a solution to this integral equation, commonly referred to as the bridge function. However, prior work has often been limited by relying on pre-specified kernel functions, which are not data adaptive and struggle to scale to large datasets. In this work, we introduce a flexible and scalable method based on a deep neural network to estimate causal effects in the presence of unmeasured confounding using proximal inference. Our method achieves state of the art performance on two well-established proximal inference benchmarks. Finally, we provide theoretical consistency guarantees for our method.

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