Posterior Probabilities: Nonmonotonicity, Asymptotic Rates, Log-Concavity, and Tur\'an's Inequality

TL;DR

This paper explores the nonmonotonic behavior of Bayesian posterior probabilities under data generated from different parameters, analyzing their asymptotic rates, log-concavity, and connections to Turán's inequality.

Contribution

It provides new insights into the nonmonotonicity and asymptotic properties of posterior probabilities, including log-concavity and specific inequalities, in the context of exponential families.

Findings

Posterior probabilities can oscillate when data are generated under different parameters.

Asymptotic rates show the expectation of the posterior is eventually decreasing.

In Bernoulli cases, a Turán-type inequality relates to the log-concavity of the posterior expectation.

Abstract

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.