A generalized back-door criterion

Abstract

We generalize Pearl's back-door criterion for directed acyclic graphs (DAGs) to more general types of graphs that describe Markov equivalence classes of DAGs and/or allow for arbitrarily many hidden variables. We also give easily checkable necessary and sufficient graphical criteria for the existence of a set of variables that satisfies our generalized back-door criterion, when considering a single intervention and a single outcome variable. Moreover, if such a set exists, we provide an explicit set that fulfills the criterion. We illustrate the results in several examples. R-code is available in the R-package pcalg.1. Introduction. Causal Bayesian networks are widely used for causal reasoning [e.g., Glymour et al. (1987), Koller and Friedman (2009), Pearl (1995, 2000, 2009, Scheines (1993, 2000)]. In particular, if the causal structure is known and represented by a directed acyclic graph (DAG), this framework allows one to deduce post-intervention distributions and causal effects from the pre-intervention (or observational) distribution. Hence, if the causal DAG is known, one can estimate causal effects from observational data. Covariate adjustment is often used for this purpose. The back-door criterion [Pearl (1993)] is a graphical criterion that is sufficient for adjustment, in the sense that a set of variables can be used for covariate adjustment if it satisfies the back-door criterion for the given graph.In practice, there are two important complications. First, the underlying DAG may be unknown. In this case one can try to estimate the DAG, but in general one cannot identify the underlying DAG uniquely. Instead, one can identify its Markov equivalence class, which consists of all DAGs that encode the same conditional independence relationships as the underlying DAG. Such a Markov equivalence class can be represented uniquely

Estimating bounds on causal effects in high-dimensional and possibly confounded systems

“…possibleDe ( X, 𝒢) and De ( X, 𝒢) are equal if 𝒢 is a MAG. Maathuis and Colombo (2015) prove the following theorem:…”

“…3). Maathuis and Colombo (2015) extend Pearl’s back-door criterion for DAGs to the graphical structures above: CPDAGs, MAGs, and PAGs. The sufficient back-door set is more complicated but the principle is the same.…”

“…All of these definitions can be found in Maathuis and Colombo (2015); the definition of visible/invisible edges is a generalization of the standard one introduced in Zhang (2008a). A visible edge between X and Y in a MAG or PAG picks out an ancestral relationship that is incompatible with any latent common cause between X and Y in the underlying DAG.…”

“…The causal effects of some variables may not be identifiable by Maathuis and Colombo’s generalized back-door criterion, as is clear from the definition. They may sometimes be identifiable by other means (Maathuis and Colombo, 2015; Perković et al, 2015; Hyttinen et al, 2015). When an LV-IDA estimate is “NA” this indicates that the measured set of covariates is not sufficient to rule out (using the back-door criterion) confounding in some MAG consistent with the data.…”

“…The back-door criterion is a graphical criterion that is sufficient for adjustment in the following sense: if a set of variables satisfies the back-door criterion for a given graph, then conditioning on that set is sufficient for estimating intervention effects from observed distributions. Maathuis and Colombo (2015) generalize the back-door criterion to different types of graphical objects, and their result will play an instrumental role in the algorithms we propose. In order to estimate intervention effects via (generalized) back-door adjustment from data, the researcher must be able to identify the set of covariates which satisfy the (generalized) back-door criterion.…”