Ilya Shpitser

Razieh Nabi, Rohit Bhattacharya, Ilya Shpitser et al.

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.

Yuqin Yang, AmirEmad Ghassami, Mohamed Nafea et al.

We focus on causal discovery in the presence of measurement error in linear systems where the mixing matrix, i.e., the matrix indicating the independent exogenous noise terms pertaining to the observed variables, is identified up to permutation and scaling of the columns. We demonstrate a somewhat surprising connection between this problem and causal discovery in the presence of unobserved parentless causes, in the sense that there is a mapping, given by the mixing matrix, between the underlying models to be inferred in these problems. Consequently, any identifiability result based on the mixing matrix for one model translates to an identifiability result for the other model. We characterize to what extent the causal models can be identified under a two-part faithfulness assumption. Under only the first part of the assumption (corresponding to the conventional definition of faithfulness), the structure can be learned up to the causal ordering among an ordered grouping of the variables but not all the edges across the groups can be identified. We further show that if both parts of the faithfulness assumption are imposed, the structure can be learned up to a more refined ordered grouping. As a result of this refinement, for the latent variable model with unobserved parentless causes, the structure can be identified. Based on our theoretical results, we propose causal structure learning methods for both models, and evaluate their performance on synthetic data.

Ilya Shpitser

The ID algorithm solves the problem of identification of interventional distributions of the form p(Y | do(a)) in graphical causal models, and has been formulated in a number of ways [12, 9, 6]. The ID algorithm is sound (outputs the correct functional of the observed data distribution whenever p(Y | do(a)) is identified in the causal model represented by the input graph), and complete (explicitly flags as a failure any input p(Y | do(a)) whenever this distribution is not identified in the causal model represented by the input graph). The reference [9] provides a result, the so called "hedge criterion" (Corollary 3), which aims to give a graphical characterization of situations when the ID algorithm fails to identify its input in terms of a structure in the input graph called the hedge. While the ID algorithm is, indeed, a sound and complete algorithm, and the hedge structure does arise whenever the input distribution is not identified, Corollary 3 presented in [9] is incorrect as stated. In this note, I outline the modern presentation of the ID algorithm, discuss a simple counterexample to Corollary 3, and provide a number of graphical characterizations of the ID algorithm failing to identify its input distribution.

Zixiao Wang, AmirEmad Ghassami, Ilya Shpitser

We consider the task of identifying and estimating a parameter of interest in settings where data is missing not at random (MNAR). In general, such parameters are not identified without strong assumptions on the missing data model. In this paper, we take an alternative approach and introduce a method inspired by data fusion, where information in an MNAR dataset is augmented by information in an auxiliary dataset subject to missingness at random (MAR). We show that even if the parameter of interest cannot be identified given either dataset alone, it can be identified given pooled data, under two complementary sets of assumptions. We derive an inverse probability weighted (IPW) estimator for identified parameters, and evaluate the performance of our estimation strategies via simulation studies, and a data application.

Rohit Bhattacharya, Tushar Nagarajan, Daniel Malinsky et al.

The data drawn from biological, economic, and social systems are often confounded due to the presence of unmeasured variables. Prior work in causal discovery has focused on discrete search procedures for selecting acyclic directed mixed graphs (ADMGs), specifically ancestral ADMGs, that encode ordinary conditional independence constraints among the observed variables of the system. However, confounded systems also exhibit more general equality restrictions that cannot be represented via these graphs, placing a limit on the kinds of structures that can be learned using ancestral ADMGs. In this work, we derive differentiable algebraic constraints that fully characterize the space of ancestral ADMGs, as well as more general classes of ADMGs, arid ADMGs and bow-free ADMGs, that capture all equality restrictions on the observed variables. We use these constraints to cast causal discovery as a continuous optimization problem and design differentiable procedures to find the best fitting ADMG when the data comes from a confounded linear system of equations with correlated errors. We demonstrate the efficacy of our method through simulations and application to a protein expression dataset. Code implementing our methods is open-source and publicly available at https://gitlab.com/rbhatta8/dcd and will be incorporated into the Ananke package.

Henrik von Kleist, Alireza Zamanian, Ilya Shpitser et al.

Active feature acquisition (AFA) agents, crucial in domains like healthcare where acquiring features is often costly or harmful, determine the optimal set of features for a subsequent classification task. As deploying an AFA agent introduces a shift in missingness distribution, it's vital to assess its expected performance at deployment using retrospective data. In a companion paper, we introduce a semi-offline reinforcement learning (RL) framework for active feature acquisition performance evaluation (AFAPE) where features are assumed to be time-dependent. Here, we study and extend the AFAPE problem to cover static feature settings, where features are time-invariant, and hence provide more flexibility to the AFA agents in deciding the order of the acquisitions. In this static feature setting, we derive and adapt new inverse probability weighting (IPW), direct method (DM), and double reinforcement learning (DRL) estimators within the semi-offline RL framework. These estimators can be applied when the missingness in the retrospective dataset follows a missing-at-random (MAR) pattern. They also can be applied to missing-not-at-random (MNAR) patterns in conjunction with appropriate existing missing data techniques. We illustrate the improved data efficiency offered by the semi-offline RL estimators in synthetic and real-world data experiments under synthetic MAR and MNAR missingness.

Trung Phung, Kyle Reese, Ilya Shpitser et al.

A common approach for handling missing values in data analysis pipelines is multiple imputation via software packages such as MICE (Van Buuren and Groothuis-Oudshoorn, 2011) and Amelia (Honaker et al., 2011). These packages typically assume the data are missing at random (MAR), and impose parametric or smoothing assumptions upon the imputing distributions in a way that allows imputation to proceed even if not all missingness patterns have support in the data. Such assumptions are unrealistic in practice, and induce model misspecification bias on any analysis performed after such imputation. In this paper, we provide a principled alternative. Specifically, we develop a new characterization for the full data law in graphical models of missing data. This characterization is constructive, is easily adapted for the calculation of imputation distributions for both MAR and MNAR (missing not at random) mechanisms, and is able to handle lack of support for certain patterns of missingness. We use this characterization to develop a new imputation algorithm -- Multivariate Imputation via Supported Pattern Recursion (MISPR) -- which uses Gibbs sampling, by analogy with the Multivariate Imputation with Chained Equations (MICE) algorithm, but which is consistent under both MAR and MNAR settings, and is able to handle missing data patterns with no support without imposing additional assumptions beyond those already imposed by the missing data model itself. In simulations, we show MISPR obtains comparable results to MICE when data are MAR, and superior, less biased results when data are MNAR. Our characterization and imputation algorithm based on it are a step towards making principled missing data methods more practical in applied settings, where the data are likely both MNAR and sufficiently high dimensional to yield missing data patterns with no support at available sample sizes.

Helen Guo, Elizabeth L. Ogburn, Ilya Shpitser

Identifying causal effects in the presence of unmeasured variables is a fundamental challenge in causal inference, for which proxy variable methods have emerged as a powerful solution. We contrast two major approaches in this framework: (1) bridge equation methods, which leverage solutions to integral equations to recover causal targets, and (2) array decomposition methods, which recover latent factors composing counterfactual quantities by exploiting unique determination of eigenspaces. We compare the model restrictions underlying these two approaches and provide insight into implications of the underlying assumptions, clarifying the scope of applicability for each method.

Razieh Nabi, Rohit Bhattacharya, Ilya Shpitser et al.

We are grateful to the discussants, Levis and Kennedy [2025], Luo and Geng [2025], Wang and van der Laan [2025], and Yang and Kim [2025], for their thoughtful comments on our paper (Nabi et al., 2025). In this rejoinder, we summarize our main contributions and respond to each discussion in turn.

Trung Phung, Jaron J. R. Lee, Opeyemi Oladapo-Shittu et al.

A common type of zero-inflated data has certain true values incorrectly replaced by zeros due to data recording conventions (rare outcomes assumed to be absent) or details of data recording equipment (e.g. artificial zeros in gene expression data). Existing methods for zero-inflated data either fit the observed data likelihood via parametric mixture models that explicitly represent excess zeros, or aim to replace excess zeros by imputed values. If the goal of the analysis relies on knowing true data realizations, a particular challenge with zero-inflated data is identifiability, since it is difficult to correctly determine which observed zeros are real and which are inflated. This paper views zero-inflated data as a general type of missing data problem, where the observability indicator for a potentially censored variable is itself unobserved whenever a zero is recorded. We show that, without additional assumptions, target parameters involving a zero-inflated variable are not identified. However, if a proxy of the missingness indicator is observed, a modification of the effect restoration approach of Kuroki and Pearl allows identification and estimation, given the proxy-indicator relationship is known. If this relationship is unknown, our approach yields a partial identification strategy for sensitivity analysis. Specifically, we show that only certain proxy-indicator relationships are compatible with the observed data distribution. We give an analytic bound for this relationship in cases with a categorical outcome, which is sharp in certain models. For more complex cases, sharp numerical bounds may be computed using methods in Duarte et al.[2023]. We illustrate our method via simulation studies and a data application on central line-associated bloodstream infections (CLABSIs).

AmirEmad Ghassami, Chang Liu, Alan Yang et al.

We study identifying and estimating the causal effect of a treatment variable on a long-term outcome using data from an observational and an experimental domain. The observational data are subject to unobserved confounding. Furthermore, subjects in the experiment are only followed for a short period; thus, long-term effects are unobserved, though short-term effects are available. Consequently, neither data source alone suffices for causal inference on the long-term outcome, necessitating a principled fusion of the two. We propose three approaches for data fusion for the purpose of identifying and estimating the causal effect. The first assumes equal confounding bias for short-term and long-term outcomes. The second weakens this assumption by leveraging an observed confounder for which the short-term and long-term potential outcomes share the same partial additive association with this confounder. The third approach employs proxy variables of the latent confounder of the treatment-outcome relationship, extending the proximal causal inference framework to the data fusion setting. For each approach, we develop influence function-based estimators and analyze their robustness properties. We illustrate our methods by estimating the effect of class size on 8th-grade SAT scores using data from the Project STAR experiment combined with observational data from the Early Childhood Longitudinal Study.

Nathan Drenkow, Numair Sani, Ilya Shpitser et al.

Deep neural networks for computer vision are deployed in increasingly safety-critical and socially-impactful applications, motivating the need to close the gap in model performance under varied, naturally occurring imaging conditions. Robustness, ambiguously used in multiple contexts including adversarial machine learning, refers here to preserving model performance under naturally-induced image corruptions or alterations. We perform a systematic review to identify, analyze, and summarize current definitions and progress towards non-adversarial robustness in deep learning for computer vision. We find this area of research has received disproportionately less attention relative to adversarial machine learning, yet a significant robustness gap exists that manifests in performance degradation similar in magnitude to adversarial conditions. Toward developing a more transparent definition of robustness, we provide a conceptual framework based on a structural causal model of the data generating process and interpret non-adversarial robustness as pertaining to a model's behavior on corrupted images corresponding to low-probability samples from the unaltered data distribution. We identify key architecture-, data augmentation-, and optimization tactics for improving neural network robustness. This robustness perspective reveals that common practices in the literature correspond to causal concepts. We offer perspectives on how future research may mind this evident and significant non-adversarial robustness gap.

AmirEmad Ghassami, Alan Yang, Ilya Shpitser et al.

Proximal causal inference was recently proposed as a framework to identify causal effects from observational data in the presence of hidden confounders for which proxies are available. In this paper, we extend the proximal causal inference approach to settings where identification of causal effects hinges upon a set of mediators which are not observed, yet error prone proxies of the hidden mediators are measured. Specifically, (i) We establish causal hidden mediation analysis, which extends classical causal mediation analysis methods for identifying natural direct and indirect effects under no unmeasured confounding to a setting where the mediator of interest is hidden, but proxies of it are available. (ii) We establish hidden front-door criterion, which extends the classical front-door criterion to allow for hidden mediators for which proxies are available. (iii) We show that the identification of a certain causal effect called population intervention indirect effect remains possible with hidden mediators in settings where challenges in (i) and (ii) might co-exist. We view (i)-(iii) as important steps towards the practical application of front-door criteria and mediation analysis as mediators are almost always measured with error and thus, the most one can hope for in practice is that the measurements are at best proxies of mediating mechanisms. We propose identification approaches for the parameters of interest in our considered models. For the estimation aspect, we propose an influence function-based estimation method and provide an analysis for the robustness of the estimators.

AmirEmad Ghassami, Ilya Shpitser

Graphical causal models led to the development of complete non-parametric identification theory in arbitrary structured systems, and general approaches to efficient inference. Nevertheless, graphical approaches to causal inference have not been embraced by the statistics and public health communities. In those communities causal assumptions are instead expressed in terms of potential outcomes, or responses to hypothetical interventions. Such interventions are generally conceptualized only on a limited set of variables, where the corresponding experiment could, in principle, be performed. By contrast, graphical approaches to causal inference generally assume interventions on all variables are well defined - an overly restrictive and unrealistic assumption that may have limited the adoption of these approaches in applied work in statistics and public health. In this paper, we build on a unification of graphical and potential outcomes approaches to causality exemplified by Single World Intervention Graphs (SWIGs) to define graphical models with a restricted set of allowed interventions. We give a complete identification theory for such models, and develop a complete calculus of interventions based on a generalization of the do-calculus, and axioms that govern probabilistic operations on Markov kernels. A corollary of our results is a complete identification theory for causal effects in another graphical framework with a restricted set of interventions, the decision theoretic graphical formulation of causality.

Guilherme Duarte, Noam Finkelstein, Dean Knox et al.

When causal quantities cannot be point identified, researchers often pursue partial identification to quantify the range of possible values. However, the peculiarities of applied research conditions can make this analytically intractable. We present a general and automated approach to causal inference in discrete settings. We show causal questions with discrete data reduce to polynomial programming problems, and we present an algorithm to automatically bound causal effects using efficient dual relaxation and spatial branch-and-bound techniques. The user declares an estimand, states assumptions, and provides data (however incomplete or mismeasured). The algorithm then searches over admissible data-generating processes and outputs the most precise possible range consistent with available information -- i.e., sharp bounds -- including a point-identified solution if one exists. Because this search can be computationally intensive, our procedure reports and continually refines non-sharp ranges that are guaranteed to contain the truth at all times, even when the algorithm is not run to completion. Moreover, it offers an additional guarantee we refer to as $ε$-sharpness, characterizing the worst-case looseness of the incomplete bounds. Analytically validated simulations show the algorithm accommodates classic obstacles, including confounding, selection, measurement error, noncompliance, and nonresponse.

Ilya Shpitser, Zach Wood-Doughty, Eric J. Tchetgen Tchetgen

Unobserved confounding is a fundamental obstacle to establishing valid causal conclusions from observational data. Two complementary types of approaches have been developed to address this obstacle: obtaining identification using fortuitous external aids, such as instrumental variables or proxies, or by means of the ID algorithm, using Markov restrictions on the full data distribution encoded in graphical causal models. In this paper we aim to develop a synthesis of the former and latter approaches to identification in causal inference to yield the most general identification algorithm in multivariate systems currently known -- the proximal ID algorithm. In addition to being able to obtain nonparametric identification in all cases where the ID algorithm succeeds, our approach allows us to systematically exploit proxies to adjust for the presence of unobserved confounders that would have otherwise prevented identification. In addition, we outline a class of estimation strategies for causal parameters identified by our method in an important special case. We illustrate our approach by simulation studies and a data application.

Noam Finkelstein, Beata Zjawin, Elie Wolfe et al.

Directed acyclic graphs (DAGs) with hidden variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. In this work, we present entropic inequality constraints that are implied by $e$-separation relations in hidden variable DAGs with discrete observed variables. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy; e.g. at the extreme case, a variable with entropy $0$ can convey no information. We show how these constraints can be used to learn about the true causal model from an observed data distribution. In addition, we propose a measure of causal influence called the minimal mediary entropy, and demonstrate that it can augment traditional measures such as the average causal effect.

Yizhen Xu, Numair Sani, AmirEmad Ghassami et al.

In many applications, researchers are interested in the direct and indirect causal effects of a treatment or exposure on an outcome of interest. Mediation analysis offers a rigorous framework for identifying and estimating these causal effects. For binary treatments, efficient estimators for the direct and indirect effects are presented by Tchetgen Tchetgen and Shpitser (2012) based on the influence function of the parameter of interest. These estimators possess desirable properties such as multiple-robustness and asymptotic normality while allowing for slower than root-n rates of convergence for the nuisance parameters. However, in settings involving continuous treatments, these influence function-based estimators are not readily applicable without making strong parametric assumptions. In this work, utilizing a kernel-smoothing approach, we propose an estimator suitable for settings with continuous treatments inspired by the influence function-based estimator of Tchetgen Tchetgen and Shpitser (2012). Our proposed approach employs cross-fitting, relaxing the smoothness requirements on the nuisance functions and allowing them to be estimated at slower rates than the target parameter. Additionally, similar to influence function-based estimators, our proposed estimator is multiply robust and asymptotically normal, allowing for inference in settings where parametric assumptions may not be justified.

AmirEmad Ghassami, Andrew Ying, Ilya Shpitser et al.

Robins et al. (2008) introduced a class of influence functions (IFs) which could be used to obtain doubly robust moment functions for the corresponding parameters. However, that class does not include the IF of parameters for which the nuisance functions are solutions to integral equations. Such parameters are particularly important in the field of causal inference, specifically in the recently proposed proximal causal inference framework of Tchetgen Tchetgen et al. (2020), which allows for estimating the causal effect in the presence of latent confounders. In this paper, we first extend the class of Robins et al. to include doubly robust IFs in which the nuisance functions are solutions to integral equations. Then we demonstrate that the double robustness property of these IFs can be leveraged to construct estimating equations for the nuisance functions, which enables us to solve the integral equations without resorting to parametric models. We frame the estimation of the nuisance functions as a minimax optimization problem. We provide convergence rates for the nuisance functions and conditions required for asymptotic linearity of the estimator of the parameter of interest. The experiment results demonstrate that our proposed methodology leads to robust and high-performance estimators for average causal effect in the proximal causal inference framework.

Zach Wood-Doughty, Ilya Shpitser, Mark Dredze

Drawing causal conclusions from observational data requires making assumptions about the true data-generating process. Causal inference research typically considers low-dimensional data, such as categorical or numerical fields in structured medical records. High-dimensional and unstructured data such as natural language complicates the evaluation of causal inference methods; such evaluations rely on synthetic datasets with known causal effects. Models for natural language generation have been widely studied and perform well empirically. However, existing methods not immediately applicable to producing synthetic datasets for causal evaluations, as they do not allow for quantifying a causal effect on the text itself. In this work, we develop a framework for adapting existing generation models to produce synthetic text datasets with known causal effects. We use this framework to perform an empirical comparison of four recently-proposed methods for estimating causal effects from text data. We release our code and synthetic datasets.

Noam Finkelstein, Roy Adams, Suchi Saria et al.

When data contains measurement errors, it is necessary to make assumptions relating the observed, erroneous data to the unobserved true phenomena of interest. These assumptions should be justifiable on substantive grounds, but are often motivated by mathematical convenience, for the sake of exactly identifying the target of inference. We adopt the view that it is preferable to present bounds under justifiable assumptions than to pursue exact identification under dubious ones. To that end, we demonstrate how a broad class of modeling assumptions involving discrete variables, including common measurement error and conditional independence assumptions, can be expressed as linear constraints on the parameters of the model. We then use linear programming techniques to produce sharp bounds for factual and counterfactual distributions under measurement error in such models. We additionally propose a procedure for obtaining outer bounds on non-linear models. Our method yields sharp bounds in a number of important settings -- such as the instrumental variable scenario with measurement error -- for which no bounds were previously known.

Ranjani Srinivasan, Jaron Lee, Rohit Bhattacharya et al.

Causal analyses of longitudinal data generally assume that the qualitative causal structure relating variables remains invariant over time. In structured systems that transition between qualitatively different states in discrete time steps, such an approach is deficient on two fronts. First, time-varying variables may have state-specific causal relationships that need to be captured. Second, an intervention can result in state transitions downstream of the intervention different from those actually observed in the data. In other words, interventions may counterfactually alter the subsequent temporal evolution of the system. We introduce a generalization of causal graphical models, Path Dependent Structural Equation Models (PDSEMs), that can describe such systems. We show how causal inference may be performed in such models and illustrate its use in simulations and data obtained from a septoplasty surgical procedure.

Noam Finkelstein, Ilya Shpitser

Causal parameters may not be point identified in the presence of unobserved confounding. However, information about non-identified parameters, in the form of bounds, may still be recovered from the observed data in some cases. We develop a new general method for obtaining bounds on causal parameters using rules of probability and restrictions on counterfactuals implied by causal graphical models. We additionally provide inequality constraints on functionals of the observed data law implied by such causal models. Our approach is motivated by the observation that logical relations between identified and non-identified counterfactual events often yield information about non-identified events. We show that this approach is powerful enough to recover known sharp bounds and tight inequality constraints, and to derive novel bounds and constraints.

Numair Sani, Jaron Lee, Razieh Nabi et al.

Black box models in machine learning have demonstrated excellent predictive performance in complex problems and high-dimensional settings. However, their lack of transparency and interpretability restrict the applicability of such models in critical decision-making processes. In order to combat this shortcoming, we propose a novel approach to trading off interpretability and performance in prediction models using ideas from semiparametric statistics, allowing us to combine the interpretability of parametric regression models with performance of nonparametric methods. We achieve this by utilizing a two-piece model: the first piece is interpretable and parametric, to which a second, uninterpretable residual piece is added. The performance of the overall model is optimized using methods from the sufficient dimension reduction literature. Influence function based estimators are derived and shown to be doubly robust. This allows for use of approaches such as double Machine Learning in estimating our model parameters. We illustrate the utility of our approach via simulation studies and a data application based on predicting the length of stay in the intensive care unit among surgery patients.

Numair Sani, Daniel Malinsky, Ilya Shpitser

Causal approaches to post-hoc explainability for black-box prediction models (e.g., deep neural networks trained on image pixel data) have become increasingly popular. However, existing approaches have two important shortcomings: (i) the "explanatory units" are micro-level inputs into the relevant prediction model, e.g., image pixels, rather than interpretable macro-level features that are more useful for understanding how to possibly change the algorithm's behavior, and (ii) existing approaches assume there exists no unmeasured confounding between features and target model predictions, which fails to hold when the explanatory units are macro-level variables. Our focus is on the important setting where the analyst has no access to the inner workings of the target prediction algorithm, rather only the ability to query the output of the model in response to a particular input. To provide causal explanations in such a setting, we propose to learn causal graphical representations that allow for arbitrary unmeasured confounding among features. We demonstrate the resulting graph can differentiate between interpretable features that causally influence model predictions versus those that are merely associated with model predictions due to confounding. Our approach is motivated by a counterfactual theory of causal explanation wherein good explanations point to factors that are "difference-makers" in an interventionist sense.

Razieh Nabi, Rohit Bhattacharya, Ilya Shpitser

Missing data has the potential to affect analyses conducted in all fields of scientific study, including healthcare, economics, and the social sciences. Several approaches to unbiased inference in the presence of non-ignorable missingness rely on the specification of the target distribution and its missingness process as a probability distribution that factorizes with respect to a directed acyclic graph. In this paper, we address the longstanding question of the characterization of models that are identifiable within this class of missing data distributions. We provide the first completeness result in this field of study -- necessary and sufficient graphical conditions under which, the full data distribution can be recovered from the observed data distribution. We then simultaneously address issues that may arise due to the presence of both missing data and unmeasured confounding, by extending these graphical conditions and proofs of completeness, to settings where some variables are not just missing, but completely unobserved.

Eli Sherman, David Arbour, Ilya Shpitser

In many applied fields, researchers are often interested in tailoring treatments to unit-level characteristics in order to optimize an outcome of interest. Methods for identifying and estimating treatment policies are the subject of the dynamic treatment regime literature. Separately, in many settings the assumption that data are independent and identically distributed does not hold due to inter-subject dependence. The phenomenon where a subject's outcome is dependent on his neighbor's exposure is known as interference. These areas intersect in myriad real-world settings. In this paper we consider the problem of identifying optimal treatment policies in the presence of interference. Using a general representation of interference, via Lauritzen-Wermuth-Freydenburg chain graphs (Lauritzen and Richardson, 2002), we formalize a variety of policy interventions under interference and extend existing identification theory (Tian, 2008; Sherman and Shpitser, 2018). Finally, we illustrate the efficacy of policy maximization under interference in a simulation study.

Jaron J. R. Lee, Ilya Shpitser

Causal inference quantifies cause-effect relationships by estimating counterfactual parameters from data. This entails using \emph{identification theory} to establish a link between counterfactual parameters of interest and distributions from which data is available. A line of work characterized non-parametric identification for a wide variety of causal parameters in terms of the \emph{observed data distribution}. More recently, identification results have been extended to settings where experimental data from interventional distributions is also available. In this paper, we use Single World Intervention Graphs and a nested factorization of models associated with mixed graphs to give a very simple view of existing identification theory for experimental data. We use this view to yield general identification algorithms for settings where the input distributions consist of an arbitrary set of observational and experimental distributions, including marginal and conditional distributions. We show that for problems where inputs are interventional marginal distributions of a certain type (ancestral marginals), our algorithm is complete.

Rohit Bhattacharya, Razieh Nabi, Ilya Shpitser

Identification theory for causal effects in causal models associated with hidden variable directed acyclic graphs (DAGs) is well studied. However, the corresponding algorithms are underused due to the complexity of estimating the identifying functionals they output. In this work, we bridge the gap between identification and estimation of population-level causal effects involving a single treatment and a single outcome. We derive influence function based estimators that exhibit double robustness for the identified effects in a large class of hidden variable DAGs where the treatment satisfies a simple graphical criterion; this class includes models yielding the adjustment and front-door functionals as special cases. We also provide necessary and sufficient conditions under which the statistical model of a hidden variable DAG is nonparametrically saturated and implies no equality constraints on the observed data distribution. Further, we derive an important class of hidden variable DAGs that imply observed data distributions observationally equivalent (up to equality constraints) to fully observed DAGs. In these classes of DAGs, we derive estimators that achieve the semiparametric efficiency bounds for the target of interest where the treatment satisfies our graphical criterion. Finally, we provide a sound and complete identification algorithm that directly yields a weight based estimation strategy for any identifiable effect in hidden variable causal models.

Elizabeth L. Ogburn, Ilya Shpitser, Eric J. Tchetgen Tchetgen

This note has been updated (April, 2020) to respond to "Towards Clarifying the Theory of the Deconfounder" by Yixin Wang, David M. Blei (arXiv:2003.04948). This original note, posted in January, 2020, is meant to complement our previous comment on "The Blessings of Multiple Causes" by Wang and Blei (2019). We provide a more succinct and transparent explanation of the fact that the deconfounder does not control for multi-cause confounding. The argument given in Wang and Blei (2019) makes two mistakes: (1) attempting to infer independence conditional on one variable from independence conditional on a different, unrelated variable, and (2) attempting to infer joint independence from pairwise independence. We give two simple counterexamples to the deconfounder claim.

Elizabeth L. Ogburn, Ilya Shpitser, Eric J. Tchetgen Tchetgen

(This comment has been updated to respond to Wang and Blei's rejoinder [arXiv:1910.07320].) The premise of the deconfounder method proposed in "Blessings of Multiple Causes" by Wang and Blei [arXiv:1805.06826], namely that a variable that renders multiple causes conditionally independent also controls for unmeasured multi-cause confounding, is incorrect. This can be seen by noting that no fact about the observed data alone can be informative about ignorability, since ignorability is compatible with any observed data distribution. Methods to control for unmeasured confounding may be valid with additional assumptions in specific settings, but they cannot, in general, provide a checkable approach to causal inference, and they do not, in general, require weaker assumptions than the assumptions that are commonly used for causal inference. While this is outside the scope of this comment, we note that much recent work on applying ideas from latent variable modeling to causal inference problems suffers from similar issues.

Razieh Nabi, Daniel Malinsky, Ilya Shpitser

Recently there has been sustained interest in modifying prediction algorithms to satisfy fairness constraints. These constraints are typically complex nonlinear functionals of the observed data distribution. Focusing on the path-specific causal constraints proposed by Nabi and Shpitser (2018), we introduce new theoretical results and optimization techniques to make model training easier and more accurate. Specifically, we show how to reparameterize the observed data likelihood such that fairness constraints correspond directly to parameters that appear in the likelihood, transforming a complex constrained optimization objective into a simple optimization problem with box constraints. We also exploit methods from empirical likelihood theory in statistics to improve predictive performance by constraining baseline covariates, without requiring parametric models. We combine the merits of both proposals to optimize a hybrid reparameterized likelihood. The techniques presented here should be applicable more broadly to fair prediction proposals that impose constraints on predictive models.

Rohit Bhattacharya, Razieh Nabi, Ilya Shpitser et al.

Missing data is a pervasive problem in data analyses, resulting in datasets that contain censored realizations of a target distribution. Many approaches to inference on the target distribution using censored observed data, rely on missing data models represented as a factorization with respect to a directed acyclic graph. In this paper we consider the identifiability of the target distribution within this class of models, and show that the most general identification strategies proposed so far retain a significant gap in that they fail to identify a wide class of identifiable distributions. To address this gap, we propose a new algorithm that significantly generalizes the types of manipulations used in the ID algorithm, developed in the context of causal inference, in order to obtain identification.

Rohit Bhattacharya, Daniel Malinsky, Ilya Shpitser

Classical causal and statistical inference methods typically assume the observed data consists of independent realizations. However, in many applications this assumption is inappropriate due to a network of dependences between units in the data. Methods for estimating causal effects have been developed in the setting where the structure of dependence between units is known exactly, but in practice there is often substantial uncertainty about the precise network structure. This is true, for example, in trial data drawn from vulnerable communities where social ties are difficult to query directly. In this paper we combine techniques from the structure learning and interference literatures in causal inference, proposing a general method for estimating causal effects under data dependence when the structure of this dependence is not known a priori. We demonstrate the utility of our method on synthetic datasets which exhibit network dependence.

Daniel Chicharro, Stefano Panzeri, Ilya Shpitser

Constraint-based structure learning algorithms infer the causal structure of multivariate systems from observational data by determining an equivalent class of causal structures compatible with the conditional independencies in the data. Methods based on additive-noise (AN) models have been proposed to further discriminate between causal structures that are equivalent in terms of conditional independencies. These methods rely on a particular form of the generative functional equations, with an additive noise structure, which allows inferring the directionality of causation by testing the independence between the residuals of a nonlinear regression and the predictors (nrr-independencies). Full causal structure identifiability has been proven for systems that contain only additive-noise equations and have no hidden variables. We extend the AN framework in several ways. We introduce alternative regression-free tests of independence based on conditional variances (cv-independencies). We consider conditionally-additive-noise (CAN) models, in which the equations may have the AN form only after conditioning. We exploit asymmetries in nrr-independencies or cv-independencies resulting from the CAN form to derive a criterion that infers the causal relation between a pair of variables in a multivariate system without any assumption about the form of the equations or the presence of hidden variables.

Zach Wood-Doughty, Ilya Shpitser, Mark Dredze

Causal understanding is essential for many kinds of decision-making, but causal inference from observational data has typically only been applied to structured, low-dimensional datasets. While text classifiers produce low-dimensional outputs, their use in causal inference has not previously been studied. To facilitate causal analyses based on language data, we consider the role that text classifiers can play in causal inference through established modeling mechanisms from the causality literature on missing data and measurement error. We demonstrate how to conduct causal analyses using text classifiers on simulated and Yelp data, and discuss the opportunities and challenges of future work that uses text data in causal inference.

Razieh Nabi, Phyllis Kanki, Ilya Shpitser

The goal of personalized decision making is to map a unit's characteristics to an action tailored to maximize the expected outcome for that unit. Obtaining high-quality mappings of this type is the goal of the dynamic regime literature. In healthcare settings, optimizing policies with respect to a particular causal pathway may be of interest as well. For example, we may wish to maximize the chemical effect of a drug given data from an observational study where the chemical effect of the drug on the outcome is entangled with the indirect effect mediated by differential adherence. In such cases, we may wish to optimize the direct effect of a drug, while keeping the indirect effect to that of some reference treatment. [16] shows how to combine mediation analysis and dynamic treatment regime ideas to defines policies associated with causal pathways and counterfactual responses to these policies. In this paper, we derive a variety of methods for learning high quality policies of this type from data, in a causal model corresponding to a longitudinal setting of practical importance. We illustrate our methods via a dataset of HIV patients undergoing therapy, gathered in the Nigerian PEPFAR program.

Razieh Nabi, Daniel Malinsky, Ilya Shpitser

Systematic discriminatory biases present in our society influence the way data is collected and stored, the way variables are defined, and the way scientific findings are put into practice as policy. Automated decision procedures and learning algorithms applied to such data may serve to perpetuate existing injustice or unfairness in our society. In this paper, we consider how to make optimal but fair decisions, which "break the cycle of injustice" by correcting for the unfair dependence of both decisions and outcomes on sensitive features (e.g., variables that correspond to gender, race, disability, or other protected attributes). We use methods from causal inference and constrained optimization to learn optimal policies in a way that addresses multiple potential biases which afflict data analysis in sensitive contexts, extending the approach of (Nabi and Shpitser 2018). Our proposal comes equipped with the theoretical guarantee that the chosen fair policy will induce a joint distribution for new instances that satisfies given fairness constraints. We illustrate our approach with both synthetic data and real criminal justice data.

Razieh Nabi, Ilya Shpitser

In this paper, we consider the problem of fair statistical inference involving outcome variables. Examples include classification and regression problems, and estimating treatment effects in randomized trials or observational data. The issue of fairness arises in such problems where some covariates or treatments are "sensitive," in the sense of having potential of creating discrimination. In this paper, we argue that the presence of discrimination can be formalized in a sensible way as the presence of an effect of a sensitive covariate on the outcome along certain causal pathways, a view which generalizes (Pearl, 2009). A fair outcome model can then be learned by solving a constrained optimization problem. We discuss a number of complications that arise in classical statistical inference due to this view and provide workarounds based on recent work in causal and semi-parametric inference.

Ilya Shpitser, Robin J. Evans, Thomas S. Richardson et al.

Hidden variables are ubiquitous in practical data analysis, and therefore modeling marginal densities and doing inference with the resulting models is an important problem in statistics, machine learning, and causal inference. Recently, a new type of graphical model, called the nested Markov model, was developed which captures equality constraints found in marginals of directed acyclic graph (DAG) models. Some of these constraints, such as the so called `Verma constraint', strictly generalize conditional independence. To make modeling and inference with nested Markov models practical, it is necessary to limit the number of parameters in the model, while still correctly capturing the constraints in the marginal of a DAG model. Placing such limits is similar in spirit to sparsity methods for undirected graphical models, and regression models. In this paper, we give a log-linear parameterization which allows sparse modeling with nested Markov models. We illustrate the advantages of this parameterization with a simulation study.

Tyler J. VanderWeele, Ilya Shpitser

The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The literature has not, however, come to any consensus on a formal definition of a confounder, as it has given priority to the concept of confounding over that of a confounder. We consider a number of candidate definitions arising from various more informal statements made in the literature. We consider the properties satisfied by each candidate definition, principally focusing on (i) whether under the candidate definition control for all "confounders" suffices to control for "confounding" and (ii) whether each confounder in some context helps eliminate or reduce confounding bias. Several of the candidate definitions do not have these two properties. Only one candidate definition of those considered satisfies both properties. We propose that a "confounder" be defined as a pre-exposure covariate C for which there exists a set of other covariates X such that effect of the exposure on the outcome is unconfounded conditional on (X,C) but such that for no proper subset of (X,C) is the effect of the exposure on the outcome unconfounded given the subset. We also provide a conditional analogue of the above definition; and we propose a variable that helps reduce bias but not eliminate bias be referred to as a "surrogate confounder." These definitions are closely related to those given by Robins and Morgenstern [Comput. Math. Appl. 14 (1987) 869-916]. The implications that hold among the various candidate definitions are discussed.

Ilya Shpitser, Thomas S. Richardson, James M. Robins et al.

The constraints arising from DAG models with latent variables can be naturally represented by means of acyclic directed mixed graphs (ADMGs). Such graphs contain directed and bidirected arrows, and contain no directed cycles. DAGs with latent variables imply independence constraints in the distribution resulting from a 'fixing' operation, in which a joint distribution is divided by a conditional. This operation generalizes marginalizing and conditioning. Some of these constraints correspond to identifiable 'dormant' independence constraints, with the well known 'Verma constraint' as one example. Recently, models defined by a set of the constraints arising after fixing from a DAG with latents, were characterized via a recursive factorization and a nested Markov property. In addition, a parameterization was given in the discrete case. In this paper we use this parameterization to describe a parameter fitting algorithm, and a search and score structure learning algorithm for these nested Markov models. We apply our algorithms to a variety of datasets.

Ilya Shpitser, Judea Pearl

The subject of this paper is the elucidation of effects of actions from causal assumptions represented as a directed graph, and statistical knowledge given as a probability distribution. In particular, we are interested in predicting conditional distributions resulting from performing an action on a set of variables and, subsequently, taking measurements of another set. We provide a necessary and sufficient graphical condition for the cases where such distributions can be uniquely computed from the available information, as well as an algorithm which performs this computation whenever the condition holds. Furthermore, we use our results to prove completeness of do-calculus [Pearl, 1995] for the same identification problem.

Ilya Shpitser, Judea Pearl

Counterfactual statements, e.g., "my headache would be gone had I taken an aspirin" are central to scientific discourse, and are formally interpreted as statements derived from "alternative worlds". However, since they invoke hypothetical states of affairs, often incompatible with what is actually known or observed, testing counterfactuals is fraught with conceptual and practical difficulties. In this paper, we provide a complete characterization of "testable counterfactuals," namely, counterfactual statements whose probabilities can be inferred from physical experiments. We provide complete procedures for discerning whether a given counterfactual is testable and, if so, expressing its probability in terms of experimental data.

Ilya Shpitser, Judea Pearl

Many applications of causal analysis call for assessing, retrospectively, the effect of withholding an action that has in fact been implemented. This counterfactual quantity, sometimes called "effect of treatment on the treated," (ETT) have been used to to evaluate educational programs, critic public policies, and justify individual decision making. In this paper we explore the conditions under which ETT can be estimated from (i.e., identified in) experimental and/or observational studies. We show that, when the action invokes a singleton variable, the conditions for ETT identification have simple characterizations in terms of causal diagrams. We further give a graphical characterization of the conditions under which the effects of multiple treatments on the treated can be identified, as well as ways in which the ETT estimand can be constructed from both interventional and observational distributions.

Ilya Shpitser, Tyler VanderWeele, James M. Robins

Identifying effects of actions (treatments) on outcome variables from observational data and causal assumptions is a fundamental problem in causal inference. This identification is made difficult by the presence of confounders which can be related to both treatment and outcome variables. Confounders are often handled, both in theory and in practice, by adjusting for covariates, in other words considering outcomes conditioned on treatment and covariate values, weighed by probability of observing those covariate values. In this paper, we give a complete graphical criterion for covariate adjustment, which we term the adjustment criterion, and derive some interesting corollaries of the completeness of this criterion.

Ilya Shpitser, Thomas S. Richardson, James M. Robins

Probabilistic inference in graphical models is the task of computing marginal and conditional densities of interest from a factorized representation of a joint probability distribution. Inference algorithms such as variable elimination and belief propagation take advantage of constraints embedded in this factorization to compute such densities efficiently. In this paper, we propose an algorithm which computes interventional distributions in latent variable causal models represented by acyclic directed mixed graphs(ADMGs). To compute these distributions efficiently, we take advantage of a recursive factorization which generalizes the usual Markov factorization for DAGs and the more recent factorization for ADMGs. Our algorithm can be viewed as a generalization of variable elimination to the mixed graph case. We show our algorithm is exponential in the mixed graph generalization of treewidth.