Doubly Robust Criterion for Causal Inference

The semiparametric estimation approach, which includes inverse-probability-weighted and doubly robust estimation using propensity scores, is a standard tool for marginal structural models basically used in causal inference, and is rapidly being extended and generalized in various directions. On the other hand, although model selection is indispensable in statistical analysis, information criterion for selecting an appropriate marginal structure has just started to be developed. In this paper, based on the original idea of the information criterion, we derive an AIC-type criterion. We define a risk function based on the Kullback-Leibler divergence as the cornerstone of the information criterion, and treat a general causal inference model that is not necessarily of the type represented as a linear model. The causal effects to be estimated are those in the general population, such as the average treatment effect on the treated or the average treatment effect on the untreated. In light of the fact that doubly robust estimation, which allows either the model of the assignment variable or the model of the outcome variable to be wrong, is attached importance in this field, we will make the information criterion itself doubly robust, so that either one of the two can be wrong and still be a mathematically valid criterion.