Maximum Probability and Relative Entropy Maximization. Bayesian Maximum Probability and Empirical Likelihood

TL;DR

This paper surveys methods like maximum probability and relative entropy maximization, discussing their theoretical foundations, extensions, and applications to ill-posed inverse problems in a Bayesian and empirical context.

Contribution

It provides a comprehensive overview of maximum probability and Bayesian methods, including their asymptotic and empirical extensions, and discusses their probabilistic justification for inverse problems.

Findings

Probabilistic interpretation of maximum probability methods

Extensions to empirical maximum entropy and likelihood

Application to ill-posed inverse problems

Abstract

Works, briefly surveyed here, are concerned with two basic methods: Maximum Probability and Bayesian Maximum Probability; as well as with their asymptotic instances: Relative Entropy Maximization and Maximum Non-parametric Likelihood. Parametric and empirical extensions of the latter methods - Empirical Maximum Maximum Entropy and Empirical Likelihood - are also mentioned. The methods are viewed as tools for solving certain ill-posed inverse problems, called Pi-problem, Phi-problem, respectively. Within the two classes of problems, probabilistic justification and interpretation of the respective methods are discussed.