General Robust Bayes Pseudo-Posterior: Exponential Convergence results with Applications

Although Bayesian inference is an immensely popular paradigm among a large segment of scientists including statisticians, most of the applications consider the objective priors and need critical investigations (Efron, 2013, Science). And although it has several optimal properties, one major drawback of Bayesian inference is the lack of robustness against data contamination and model misspecification, which becomes pernicious in the use of objective priors. This paper presents the general formulation of a Bayes pseudo-posterior distribution yielding robust inference. Exponential convergence results related to the new pseudo-posterior and the corresponding Bayes estimators are established under the general parametric set-up and illustrations are provided for the independent stationary models and the independent non-homogenous models. For the first case, the discrete priors and the corresponding maximum posterior estimators are discussed with additional details. We further apply this new pseudo-posterior to propose robust versions of the Bayes predictive density estimators and the expected Bayes estimator for the fixed-design (normal) linear regression models; their properties are illustrated both theoretically as well as empirically.