Uniform Approximation of Bayes Solutions and Posteriors: Frequentistly Valid Bayes Inference
Nov 30 1992
The problem of deriving classically acceptable Bayesian estimation procedures is important for synthesis and reconciliation of the classical and Bayesian approaches to inference problems. In this work, we consider inference problems for location parameters. The idea is that if one can produce priors for which the posterior densities are uniformly close to thelikelihood function, then the corresponding Bayesian inference should also be clsoe to classical inference, at least for location parameters. We describe a large family of prior distributions meeting this goal.
Apart from obtaining approximations for the posterior density itself, we also derive uniform approximations to the Bayes rule and the posterior expected loss. We also demonstrate that for the priors, the samping distributions of the Bayes rul and the classical unbiased estimate are close uniformly in the parameter and that all 100(1-α_% Bayesian HPD sets have classical coverage probabilities uniformly close to 1 - α as well. All of our results are nonasymptotic in nature.