Sergiu Hart / papers /
Posterior Probabilities: Nonmonotonicity, Asymptotic Rates,
Log-Concavity, and Turan's
Posterior Probabilities: Nonmonotonicity, Asymptotic Rates, Log-Concavity, and
Sergiu Hart and Yosef Rinott
In the standard Bayesian framework data are assumed to
be generated by a distribution parametrized by θ in a parameter
over which a prior distribution π is given.
A Bayesian statistician quantifies the belief that the true parameter
is θ0 in Θ
by its posterior probability given the observed data.
We investigate the behavior of the posterior belief
in θ0 when the data are generated under some
parameter θ1, which may or may not be be the same
as θ0. 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 θ0-posterior is monotonically increasing
(i.e., it is a submartingale) when the data are generated
under that same θ0,
it need not be monotonically decreasing in general,
not even in terms of its overall expectation,
when the data are generated under a different θ1.
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
under θ1≠θ0 is eventually strictly
decreasing. Finally, we show that in a number of interesting cases this
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 Turán's inequality for Legendre polynomials.
First version: September 2019
The Hebrew University of Jerusalem, Center for Rationality DP-736,
Revised, April 2021
© Sergiu Hart