ASYMPTOTICS OF THE EXPECTED POSTERIOR

BERTRAND CLARKE^{1} AND DONGCHU SUN^{2}

1Department of Statistics, University of British Columbia, 6356 Agricultural Road, Vancouver, Canada V6T 1Z2

2Department of Statistics, 222 Math Science Building, University of Missouri-Columbia, Columbia, MO 65211, U.S.A.

(Received April 30, 1996; revised October 6, 1997)

Abstract. Let (X_{1}, …, X_{n}) be independently and identically distributed observations from an exponential family p_{θ} equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ_{0}, admits an asymptotic expansion in terms of the Fisher information I(θ_{0}), the prior w, and their first two derivatives. The leading term of the expansion is of the form n^{d/2}c_{1}(d, θ_{0}) and the second order term is of the form n^{d/2−1}c_{2}(d, θ_{0}, w), with an error term that is o(n^{d/2−1}). We identify the functions c_{1} and c_{2} explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ_{0}. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Key words and phrases: Expected posterior, asymptotics, relative entropy, chi-squared distance, Bayes factor.