# Gibbs posterior convergence and the thermodynamic formalism

## TL;DR

This paper develops a Bayesian inference framework using Gibbs posteriors for dynamical systems, analyzing their asymptotic behavior and establishing connections with thermodynamic formalism to enhance understanding of dependent process inference.

## Contribution

It introduces a Gibbs posterior approach for dynamical systems, characterizes its asymptotic behavior, and links Bayesian inference with thermodynamic formalism for dependent processes.

## Key findings

- Gibbs posteriors concentrate around solutions of a variational problem.
- Posterior consistency can be established for properly specified models.
- Connections between Bayesian inference and thermodynamic formalism are demonstrated.

## Abstract

In this paper we consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. This work establishes tight connections between Gibbs posterior inference and the thermodynamic formalism, which may inspire new proof techniques in the study of Bayesian posterior consistency for dependent processes.