Bayesian inference and the parametric bootstrap

TL;DR

This paper explores how the parametric bootstrap can efficiently compute Bayesian posterior distributions, linking Bayesian and frequentist methods, and providing algorithms for assessing Bayesian inference accuracy.

Contribution

It introduces simple importance sampling formulas related to deviance in exponential families and develops algorithms for evaluating Bayesian inference accuracy.

Findings

Importance sampling formulas simplify Bayesian computation.

Algorithms demonstrate Bayesian accuracy in model selection.

Connections established between Bayesian and frequentist analyses.

Abstract

The parametric bootstrap can be used for the efficient computation of Bayes posterior distributions. Importance sampling formulas take on an easy form relating to the deviance in exponential families and are particularly simple starting from Jeffreys invariant prior. Because of the i.i.d. nature of bootstrap sampling, familiar formulas describe the computational accuracy of the Bayes estimates. Besides computational methods, the theory provides a connection between Bayesian and frequentist analysis. Efficient algorithms for the frequentist accuracy of Bayesian inferences are developed and demonstrated in a model selection example.