@booklet {sergiuhart-ppnlati2020,
	title = {Posterior Probabilities: Nonmonotonicity, Log-Concavity, and Tur{\textexclamdown}n{\textquoteright}s Inequality},
	howpublished = {Discussion Papers},
	year = {2020},
	month = {7},
	abstract = {In the standard Bayesian framework the data are assumed to be generated by a distribution parametrized by {\c\ } in a parameter space , over which a prior distribution is defined. A Bayesian statistician quantifies the belief that the true parameter is {\c\ }_0 in by its posterior probability given the observed data. We investigate the behavior of the posterior belief in {\c\ }_0 when the data are generated under some parameter {\c\ }_1, which may or may not be be the same as {\c\ }_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 {\c\ }_0-posterior is monotonically increasing (i.e., it is a submartingale) when the data are generated under that same {\c\ }_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 {\c\ }_1; in fact, it may keep going up and down many times. In the framework of simple iid coin tosses, we show that under certain conditions the overall expected posterior of {\c\ }_0 eventually becomes monotonically decreasing when the data are generated under {\c\ }_1~{\c\ }_0. Moreover, we prove that when the prior is uniform this expected posterior is a log-concave function of the sample size, by developing an inequality that is related to Tur{\textexclamdown}n{\textquoteright}s inequality for Legendre polynomials.},
	url = {http://www.ma.huji.ac.il/hart/abs/post-seq.html},
	author = {Sergiu Hart, Yosef Rinott}
}
