Frequentistic approximations to Bayesian prevision of exchangeable random elements

Abstract

Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence ξ˜1,ξ˜2,…, are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution e˜n:=1n∑i=1nδξ˜i of the n past observations, while the latter is compared to the m-fold product e˜nm. Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts δe˜n and e˜nm, respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the ξ˜n's, and not under hypothetical product laws p0∞, as p0 varies among the admissible determinations of a random probability measure.