Frequentistic approximations to Bayesian prevision of exchangeable random elements

TL;DR

This paper derives bounds on the convergence of Bayesian posterior and predictive distributions for exchangeable sequences, using probabilistic distances and focusing on the actual data-generating law rather than a fixed prior.

Contribution

It introduces new bounds for Bayesian approximations in exchangeable sequences, emphasizing the law of the data rather than product measures, which is a novel approach.

Findings

Established bounds for posterior distribution convergence.

Provided bounds for predictive distribution approximation.

Focused on the law of the sequence rather than fixed priors.

Abstract

Given a sequence \xi_1, \xi_2,... of X-valued, exchangeable random elements, let q(\xi^(n)) and p_m(\xi^(n)) stand for posterior and predictive distribution, respectively, given \xi^(n) = (\xi_1,..., \xi_n). We provide an upper bound for limsup b_n d_[[X]](q(\xi^(n)), \delta_\empiricn) and limsup b_n d_[X^m](p_m(\xi^(n)), \empiricn^m), where \empiricn is the empirical measure, b_n is a suitable sequence of positive numbers increasing to +\infty, d_[[X]] and d_[X^m] denote distinguished weak probability distances on [[X]] and [X^m], respectively, with the proviso that [S] denotes the space of all probability measures on S. A characteristic feature of our work is that the aforesaid bounds are established under the law of the \xi_n's, unlike the more common literature on Bayesian consistency, where they are studied with respect to product measures (p_0)^\infty, as p_0 varies among the admissible determinations of a random probability measure.