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 distributions on post-action outcomes given a set of measurements. 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, and show applications to sequential decision making. 1

The construction of causal graphs from non-experimental data rests on a set of constraints that the graph structure imposes on all probability distributions compatible with the graph. These constraints are of two types: conditional independencies and algebraic constraints, first noted by Verma. While conditional independencies are well studied and frequently used in causal induction algorithms, Verma constraints are still poorly understood, and rarely applied. In this paper we examine a special subset of Verma constraints which are easy to understand, easy to identify and easy to apply; they arise from “dormant independencies, ” namely, conditional independencies that hold in interventional distributions. We give a complete algorithm for determining if a dormant independence between two sets of variables is entailed by the causal graph, such that this independence is identifiable, in other words if it resides in an interventional distribution that can be predicted without resorting to interventions. We further show the usefulness of dormant independencies in model testing and induction by giving an algorithm that uses constraints entailed by dormant independencies to prune extraneous edges from a given causal graph.

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. 1

This paper addresses the problem of identifying causal effects from nonexperimental data in a causal Bayesian network, i.e., a directed acyclic graph that represents causal relationships. The identifiability question asks whether it is possible to compute the probability of some set of (effect) variables given intervention on another set of (intervention) variables, in the presence of nonobservable (i.e., hidden or latent) variables. It is well known that the answer to the question depends on the structure of the causal Bayesian network, the set of observable variables, the set of effect variables, and the set of intervention variables. Our work is based on the work of Tian and Pearl [1, 2, 3] and our own work [4], and extends it. We show that the identify algorithm that Tian and Pearl define and prove sound for semi-Markovian models can be transfered to general causal graphs and is not only sound, but also complete. This result effectively solves the identifiability question for causal Bayesian networks that Pearl posed in 1995 [5], by providing a sound and complete algorithm for identifiability. 1

We consider a hierarchy of queries about causal relationships in graphical models, where each level in the hierarchy requires more detailed information than the one below. The hierarchy consists of three levels: associative relationships, derived from a joint distribution over the observable variables; cause-effect relationships, derived from distributions resulting from external interventions; and counterfactuals, derived from distributions that span multiple “parallel worlds ” and resulting from simultaneous, possibly conflicting observations and interventions. We completely characterize cases where a given causal query can be computed from information lower in the hierarchy, and provide algorithms that accomplish this computation. Specifically, we show when effects of interventions can be computed from observational studies, and when probabilities of counterfactuals can be computed from experimental studies. We also provide a graphical characterization of those queries which cannot be computed (by any method) from queries at a lower layer of the hierarchy.

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This paper highlights several areas where graphical techniques can be harnessed to address the problem of measurement errors in causal inference. In particular, the paper discusses the control of partially observable confounders in parametric and non parametric models and the computational problem of obtaining bias-free effect estimates in such models.

The study of transportability aims to identify conditions under which causal information learned from experiments can be reused in a different environment where only passive observations can be collected. The theory introduced in [Pearl and Bareinboim, 2011] (henceforth [PB, 2011]) defines formal conditions for such transfer but falls short of providing an effective procedure for deciding whether transportability is feasible for a given set of assumptions about differences between the source and target domains. This paper provides such procedure. It establishes a necessary and sufficient condition for deciding when causal effects in the target domain are estimable from both the statistical information available and the causal information transferred from the experiments. The paper further provides a complete algorithm for computing the transport formula, that is, a way of fusing experimental and observational information to synthesize an estimate of the desired causal relation.

We consider the problem of testing whether two variables should be adjacent (either due to a direct effect between them, or due to a hidden common cause) given an observational distribution, and a set of causal assumptions encoded as a causal diagram. In other words, given a set of edges in the diagram known to be true, we are interested in testing whether another edge ought to be in the diagram. In fully observable faithful models this problem can be easily solved with conditional independence tests. Latent variables make the problem significantly harder since they can imply certain non-adjacent variable pairs, namely those connected by so called inducing paths, are not independent conditioned on any set of variables. We characterize which variable pairs can be determined to be non-adjacent by a class of constraints due to dormant independence, that is conditional independence in identifiable interventional distributions. Furthermore, we show that particular operations on joint distributions, which we call truncations are sufficient for exhibiting these non-adjacencies. This suggests a causal discovery procedure taking advantage of these constraints in the latent variable case can restrict itself to truncations. 1

{eb,judea} at cs.ucla.edu We address the problem of estimating the effect of intervening on a set of variables X from experiments on a different set, Z, that is more accessible to manipulation. This problem, which we call z-identifiability, reduces to ordinary identifiability when Z = ∅ and, like the latter, can be given syntactic characterization using the do-calculus [Pearl, 1995; 2000]. We provide a graphical necessary and sufficient condition for z-identifiability for arbitrary sets X, Z, and Y (the outcomes). We further develop a complete algorithm for computing the causal effect of X on Y using information provided by experiments on Z. Finally, we use our results to prove completeness of do-calculus relative to z-identifiability, a result that does not follow from completeness relative to ordinary identifiability. 1