Sergiu Hart / papers / Posterior Probabilities: Nonmonotonicity, Asymptotic Rates, Log-Concavity, and Turan's Inequality

Posterior Probabilities: Nonmonotonicity, Asymptotic Rates, Log-Concavity, and Turán's Inequality

Sergiu Hart and Yosef Rinott


   
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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 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 θ0-posterior under θ1≠θ0 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 Turán's inequality for Legendre polynomials.
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