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

2 Department 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 (X1,....,Xn) be independently and identically distributed observations from an exponential family ptheta equipped with a smooth prior density w on a real d-dimensional parameter theta. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, theta0, admits an asymptotic expansion in terms of the Fisher information I(theta0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2c1(d, theta0) and the second order term is of the form nd/2-1 c2 (d, theta0, w), with an error term that is o(nd/2-1). We identify the functions c1 and c2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at theta0. 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.

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