Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism

TL;DR

This paper establishes exponential convergence of Bayesian posteriors in dynamical systems using thermodynamic formalism and large deviation principles, extending previous results to almost-additive loss functions.

Contribution

It introduces a novel approach employing non-additive thermodynamic formalism to analyze Bayesian posterior convergence for dynamical systems with almost-additive loss functions.

Findings

Proves exponential convergence of Bayesian posteriors for ergodic observations.

Extends existing results to almost-additive loss functions.

Utilizes thermodynamic formalism and large deviations instead of joinings.

Abstract

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in a recent paper by K. McGoff, S. Mukherjee and A. Nobel. Our approach makes use of non-additive thermodynamic formalism and large deviation properties instead of joinings.