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ASYMPTOTICS OF THE EXPECTED POSTERIOR

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

^{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 (*X*_{1},....,*X*_{n}) be independently
and identically distributed observations from an exponential
family *p*_{theta} 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, *theta*_{0},
admits an asymptotic expansion in terms of the Fisher
information *I*(*theta*_{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*, *theta*_{0}) and the second order term is of
the form *n*^{d/2-1} *c*_{2} (*d*, *theta*_{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 *theta*_{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.

**Source**
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