The generalized IFS Bayesian method and an associated variational principle covering the classical and dynamical cases

TL;DR

This paper develops a unified IFS Bayesian framework that extends classical and dynamical Bayesian methods, linking them to variational principles from thermodynamic formalism and information theory.

Contribution

It introduces a generalized Bayesian method for dynamical and non-dynamical settings, connecting posterior probabilities to variational principles and thermodynamic formalism.

Findings

Derivation of a dynamical Bayes' rule using the Ruelle operator

Connection between Bayesian updating and variational principles in thermodynamics

Examples illustrating the application of the generalized IFS Bayesian method

Abstract

We introduce a general IFS Bayesian method for getting posterior probabilities from prior probabilities, and also a generalized Bayes' rule, which will contemplate a dynamical, as well as a non-dynamical setting. Given a loss function ${l}$, we detail the prior and posterior items, their consequences and exhibit several examples. Taking $\Theta$ as the set of parameters and $Y$ as the set of data (which usually provides {random samples}), a general IFS is a measurable map $\tau:\Theta\times Y \to Y$, which can be interpreted as a family of maps $\tau_\theta:Y\to Y,\,\theta\in\Theta$. The main inspiration for the results we will get here comes from a paper by Zellner (with no dynamics), where Bayes' rule is related to a principle of minimization of {information.} We will show that our IFS Bayesian method which produces posterior probabilities (which are associated to holonomic probabilities) is related to the optimal solution of a variational principle, somehow corresponding to the pressure in Thermodynamic Formalism, and also to the principle of minimization of information in Information Theory. Among other results, we present the prior dynamical elements and we derive the corresponding posterior elements via the Ruelle operator of Thermodynamic Formalism; getting in this way a form of dynamical Bayes' rule.