Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms

We investigate the problem of estimating the causal effect of a treatment on individual subjects from observational data, this is a central problem in various application domains, including healthcare, social sciences, and online advertising. Within the Neyman Rubin potential outcomes model, we use the Kullback Leibler (KL) divergence between the estimated and true distributions as a measure of accuracy of the estimate, and we define the information rate of the Bayesian causal inference procedure as the (asymptotic equivalence class of the) expected value of the KL divergence between the estimated and true distributions as a function of the number of samples. Using Fano method, we establish a fundamental limit on the information rate that can be achieved by any Bayesian estimator, and show that this fundamental limit is independent of the selection bias in the observational data. We characterize the Bayesian priors on the potential (factual and counterfactual) outcomes that achieve the optimal information rate. As a consequence, we show that a particular class of priors that have been widely used in the causal inference literature cannot achieve the optimal information rate. On the other hand, a broader class of priors can achieve the optimal information rate. We go on to propose a prior adaptation procedure (which we call the information based empirical Bayes procedure) that optimizes the Bayesian prior by maximizing an information theoretic criterion on the recovered causal effects rather than maximizing the marginal likelihood of the observed (factual) data. Building on our analysis, we construct an information optimal Bayesian causal inference algorithm.