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May 2022 Posterior probabilities: Nonmonotonicity, asymptotic rates, log-concavity, and Turán’s inequality
Sergiu Hart, Yosef Rinott
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Bernoulli 28(2): 1461-1490 (May 2022). DOI: 10.3150/21-BEJ1398


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 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 θ1θ0 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 result by developing an inequality that is related to Turán’s inequality for Legendre polynomials.


Previous version: July 2020 (Hebrew University of Jerusalem, Center for Rationality DP-736).

We thank Marco Scarsini for many useful discussions. We are grateful to the referees, the associate editor, and the editor for many constructive comments, and in particular for encouraging us to consider exponential families and the Bernstein–von Mises Theorem.


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Sergiu Hart. Yosef Rinott. "Posterior probabilities: Nonmonotonicity, asymptotic rates, log-concavity, and Turán’s inequality." Bernoulli 28 (2) 1461 - 1490, May 2022.


Received: 1 April 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1398

Keywords: Bayesian analysis , expected posteriors , exponential families , Legendre polynomials , sequential observations , stochastic and likelihood ratio orders , Unimodality

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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