Bayesian robustness for hierarchical ε-contamination models

Abstract

Abstract In the robust Bayesian literature in order to investigate robustness with respect to the functional form of a base prior distribution π0 (in particular with respect to the shape of the prior tails) the e-contamination model of prior distributions Γ={π: π=(1−e)π0(θ|λ)+eq,q∈Q}, has been proposed. Here π0(θ|λ) is the base elicited prior, λ is a vector of fixed hyperparameters, q is a contamination belonging to some suitable class Q and e reflects the amount of error in π0(θ|λ). For the location-scale family of priors π0(θ|λ=τ-1π0[(θ−μ)/τ}], where λ=(μ,τ), it turns out that the posterior inference is highly sensitive to changes of λ. Therefore, instead of considering a single base prior π0, it is reasonable to replace π0(θ;|λ) by π0(θ|λ1)=∝π0(θ|λ1,λ2)h(dλ2), where λ1 is the ‘known’ part of λ and λ2 is the ‘uncertain’ part, with the uncertainty reflected by allowing hϵH (H being a class of probability measures). Classes considered here include those with unimodality and quantile constraints. Ranges of posterior probabilities of any set may be computed. Illustrations related to normal hypothesis testing are given.