GIBBS POSTERIOR CONVERGENCE AND THE THERMODYNAMIC FORMALISM

• 成果类型: Article
• 署名作者: McGoff, Kevin; Mukherjee, Sayan; Nobel, Andrew B.
• 署名单位: University of North Carolina; University of North Carolina Charlotte; Duke University; Duke University; Duke University; Duke University; University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1685
• 发表日期: 2022
• 页码: 461-496
• 关键词: hidden markov-models; DYNAMICAL-SYSTEMS; Nonparametric Regression; variable selection; Consistency; complexity; inference; entropy

摘要

In this paper we consider the posterior consistency of Bayesian inference procedures when the family of models consists of appropriate stochastic processes. Specifically, we suppose that one observes an unknown ergodic process and one has access to a family of models consisting of dependent processes. In this context, we consider Gibbs posterior inference, which is a loss-based generalization of standard Bayesian inference. Our main results characterize the asymptotic behavior of the Gibbs posterior distributions on the space of models. Furthermore, we show that in the case of properly specified models our convergence results may be used to establish posterior consistency. Our model processes are defined via the thermodynamic formalism for dynamical systems, and they allow for a large degree of dependence, including both Markov chains of unbounded orders and processes that are not Markov of any order. This work establishes close connections between Gibbs posterior inference and the thermodynamic formalism for dynamical systems, which we hope will lead to new questions and results in both nonparametric Bayesian analysis and the thermodynamic formalism.