Robust Bayes and Empirical Bayes Analysis with _-Contaminated Priors

For Bayesian analysis, an attractive method of modelling uncertainty in the prior distribution is through use of -contamination classes, i. e. , classes of distributions which have the form = (1 -) ₀ + q, ₀ being the base elicited prior, q being a "contamination, " and reflecting the amount of error in ₀ that is deemed possible. Classes of contaminations that are considered include (i) all possible contaminations, (ii) all symmetric, unimodal contaminations, and (iii) all contaminations such that is unimodal. Two issues in robust Bayesian analysis are studied. The first is that of determining the range of posterior probabilities of a set as ranges over the -contamination class. The second, more extensively studied, issue is that of selecting, in a data dependent fashion, a "good" prior distribution (the Type-II maximum likelihood prior) from the -contamination class, and using this prior in the subsequent analysis. Relationships and applications to empirical Bayes analysis are also discussed.