Sensitivity of some standard Bayesian estimates to prior uncertainty: a comparison

Abstract

We compare the sensitivity of robustness of three posterior estimators - namely the posterior mean, the posterior median and the posterior mode - to uncertainties in the prior distribution. Prior uncertainty is modeled by an ε-contamination class Γ = {(1 – ε)π0 + εq:qϵQ} where π0 is the initially specified unimodal prior with mode θ0, ε is the amount of uncertainty about π0 and Q = {all unimodal distributions q for which both the mode and the median are equal to θ0andq(θ0)≤h0} for fixed h0 > 0. Sensitivity of a posterior estimate is measured by its range as well as the sup of the posterior mean squared error as the prior varies over Γ. We also compare the posterior mean squared error of the ML-II mean with that of the above estimates.