Targeted Bayesian Learning

Abstract

Targeted maximum likelihood estimation (van der Laan & Rubin 2006) is a loss-based semi-parametric estimation method that yields a substitution estimator of a target parameter of the probability distribution of the data that solves the efficient influence curve estimating equation, and thereby yields a double robust locally efficient estimator of the parameter of interest, under regularity conditions. The Bayesian paradigm is concerned with including the researcher's prior uncertainty about the parameter through a prior distribution, which combined with the likelihood yields a posterior distribution for the parameter that reflects the researcher's posterior uncertainty. In this paper, we present a way to work under the Bayesian paradigm within the framework of targeted maximum likelihood estimation. In particular, we deal with the estimation of the so-called additive causal effect, but our results can be generalized to any d-dimensional parameter. For a general review of the proposed methodology, the readers referred to (van der Laan 2008, p. 178). We assess the performance of the proposed method through the asymp-totic convergence of the posterior distribution to a normal limit distribution, the variance and bias of the mean of the posterior distribution, and the coverage probability of the credible interval implied by the posterior distribution.