Sergiu Hart / papers /
Posterior Probabilities: Nonmonotonicity, Asymptotic Rates,
LogConcavity, and Turan's
Inequality
Posterior Probabilities: Nonmonotonicity, Asymptotic Rates, LogConcavity, and
Turán's Inequality
Sergiu Hart and Yosef Rinott
Abstract
In the standard Bayesian framework data are assumed to
be generated by a distribution parametrized by θ in a parameter
space Θ,
over which a prior distribution π is given.
A Bayesian statistician quantifies the belief that the true parameter
is θ_{0} in Θ
by its posterior probability given the observed data.
We investigate the behavior of the posterior belief
in θ_{0} when the data are generated under some
parameter θ_{1}, which may or may not be be the same
as θ_{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 θ_{0}posterior is monotonically increasing
(i.e., it is a submartingale) when the data are generated
under that same θ_{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 θ_{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
θ_{0}posterior
under θ_{1}≠θ_{0} is eventually strictly
decreasing. Finally, we show that in a number of interesting cases this
expectation is
a logconcave 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 DP736,
July 2020

Revised, April 2021

Bernoulli 28, 2 (May 2022), 14611490
doi.org/10.3150/21BEJ1398
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© Sergiu Hart