Efficient adjustment sets in causal graphical models with hidden variables

In this paper we consider the selection of covariate adjustment variables for off-policy evaluation (Precup et al., 2000) in single time contextual decision making problems. Specifically, we consider the choice of variables that suffice for estimating the value of a point exposure contextual policy by the method of covariate adjustment, when the available data come from a different policy. We assume a causal graphical model with, possibly, hidden variables in which at least one valid adjustment set is fully observable. The value of a policy, also known as the interventional mean, is defined asthe mean ofan outcome (reward)under the policy. In the statistics literature, a policy is referred to as a dynamic treatment regime (Robins, 1993; Murphy et al., 2001; Robins, 2004; Schulte et al., 2014). A practical application of the methods described in this paper is in the design of planned observational studies. Investigators designing such study might use the existing graphical criteria for identifying the class of candidate valid covariate adjustment sets (Pearl, 2000; Kuroki and Miyakawa, 2003; Shpitser et al., 2010), and then apply the methods described in this paper to select an adjustment set that satisfies one of three optimality criteria that we consider here. Each criterion is defined by selecting the observable adjustment set that yields the non-parametrically adjusted estimator with smallest asymptotic variance among those that control for observable adjustment sets in a given class, specifically the class of (i) all adjustment sets, (ii) all minimal adjustment sets, or (iii) all adjustment sets that have minimum cardinality.