  Sample size criteria are often expressed in terms of the concentration of the
posterior density, as controlled by some sort of error bound. Since this is
done pre-experimentally, one can regard the posterior density as a function of
the data. Thus, when a sample size criterion is formalized in terms of a
functional of the posterior, its value is a random variable. Generally, such
functionals have means under the true distribution. We give asymptotic
expressions for the expected value, under a fixed parameter, for certain types
of functionals of the posterior density in a Bayesian analysis. The generality
of our treatment permits us to choose functionals that encapsulate a variety of
inference criteria and large ranges of error bounds. Consequently, we get
simple inequalities which can be solved to give minimal sample sizes needed for
various estimation goals. In several parametric examples, we verify that our
asymptotic bounds give good approximations to the expected values of the
functionals they approximate. Also, our numerical computations suggest our
treatment gives reasonable results.
