Yosef Rinott (Hebrew University of Jerusalem)

“Monotonicity of convergence of posteriors, and Turan type inequalities”

Given a prior distribution on a space of states of nature, suppose we quantify our belief a given state is by computing its posterior probability having received a signal. If the signal arises under the same (true) state, does it always boost our belief that this is indeed the true state, and when it does, in what sense? What happens to this belief given a signal distributed according to a different state? Given a sequence of iid observations, the posterior probability of a parameter, when the data are generated according to the same parameter, converges to one, and to zero when the data are generated by another value of the parameter. We study monotonicity and unimodality properties of this convergence including stochastic orderings between prior and posterior, and monotonicity of the expected posterior. Some of the results apply to very general settings, and in others we focus on coin tossing. It turns out that there is a relation to Turan’s inequality for orthogonal polynomials, and in particular, to Legendre polynomials.