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 in Θ by its posterior probability given the observed data. We investigate the behavior of the posterior belief in when the data are generated under some parameter , which may or may not be the same as . 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 -posterior is monotonically increasing (i.e., it is a submartingale) when the data are generated under that same , it need not be monotonically decreasing in general, not even in terms of its overall expectation, when the data are generated under a different . 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 -posterior under 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.
Acknowledgments
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.
Citation
Sergiu Hart. Yosef Rinott. "Posterior probabilities: Nonmonotonicity, asymptotic rates, log-concavity, and Turán’s inequality." Bernoulli 28 (2) 1461 - 1490, May 2022. https://doi.org/10.3150/21-BEJ1398
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