This material is published in: I. Diaz Munoz, A.E. Hubbard, M.J. van der Laan (2011). "Targeted Bayesian Learning." In M.J. van der Laan and S. Rose, Targeted Learning: Causal Inference for Observational and Experimental Data, Chapter 28. New York, Springer.


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 asymptotic 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.



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