  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.
