  In the standard Bayesian framework data are assumed to be generated by a
distribution parametrized by $\theta$ in a parameter space $\Theta$, over which
a prior distribution $\pi$ is given. A Bayesian statistician quantifies the
belief that the true parameter is $\theta_{0}$ in $\Theta$ by its posterior
probability given the observed data. We investigate the behavior of the
posterior belief in $\theta_{0}$ when the data are generated under some
parameter $\theta_{1},$ which may or may not be the same as $\theta_{0}.$
Starting from stochastic orders, specifically, likelihood ratio dominance, that
obtain for resulting distributions of posteriors, we consider monotonicity
properties of the posterior probabilities as a function of the sample size when
data arrive sequentially. While the $\theta_{0}$-posterior is monotonically
increasing (i.e., it is a submartingale) when the data are generated under that
same $\theta_{0}$, it need not be monotonically decreasing in general, not even
in terms of its overall expectation, when the data are generated under a
different $\theta_{1}.$ In fact, it may keep going up and down many times, even
in simple cases such as iid coin tosses. We obtain precise asymptotic rates
when the data come from the wide class of exponential families of
distributions; these rates imply in particular that the expectation of the
$\theta_{0}$-posterior under $\theta_{1}\neq\theta_{0}$ is eventually strictly
decreasing. Finally, we show that in a number of interesting cases this
expectation is a log-concave function of the sample size, and thus unimodal. In
the Bernoulli case we obtain this by developing an inequality that is related
to Tur\'{a}n's inequality for Legendre polynomials.
