Bayesian Inference and the Principle of Maximum Entropy

TL;DR

This paper explores how Bayesian inference incorporates different types of information, including maximum entropy principles, to update probabilities and handle constraints, providing a unified view of statistical updating.

Contribution

It demonstrates that maximum entropy reasoning is a special case of Bayesian inference with entropy-constrained priors, clarifying their relationship.

Findings

Maximum entropy is a special case of Bayesian inference.

Posterior probabilities simplify near statistical equilibrium.

Constraints can be exact or approximate in updating probabilities.

Abstract

Bayes' theorem incorporates distinct types of information through the likelihood and prior. Direct observations of state variables enter the likelihood and modify posterior probabilities through consistent updating. Information in terms of expected values of state variables modify posterior probabilities by constraining prior probabilities to be consistent with the information. Constraints on the prior can be exact, limiting hypothetical frequency distributions to only those that satisfy the constraints, or be approximate, allowing residual deviations from the exact constraint to some degree of tolerance. When the model parameters and constraint tolerances are known, posterior probability follows directly from Bayes' theorem. When parameters and tolerances are unknown a prior for them must be specified. When the system is close to statistical equilibrium the computation of posterior probabilities is simplified due to the concentration of the prior on the maximum entropy hypothesis. The relationship between maximum entropy reasoning and Bayes' theorem from this point of view is that maximum entropy reasoning is a special case of Bayesian inference with a constrained entropy-favoring prior.