Entropy, Information, and the Updating of Probabilities
This work provides a foundational framework for probabilistic inference, potentially impacting all of ML/AI by unifying and extending existing methods, though it is a review and theoretical development rather than an incremental application.
This paper reviews the method of maximum entropy as a general inference framework, emphasizing its derivation and defining an epistemic notion of information based on Bayesian beliefs of rational agents. It proposes the logarithmic relative entropy as the unique tool for updating probabilities, unifying entropic and Bayesian methods into a single scheme that handles arbitrary priors and constraints.
This paper is a review of a particular approach to the method of maximum entropy as a general framework for inference. The discussion emphasizes the pragmatic elements in the derivation. An epistemic notion of information is defined in terms of its relation to the Bayesian beliefs of ideally rational agents. The method of updating from a prior to a posterior probability distribution is designed through an eliminative induction process. The logarithmic relative entropy is singled out as the unique tool for updating that (a) is of universal applicability; (b) that recognizes the value of prior information; and (c) that recognizes the privileged role played by the notion of independence in science. The resulting framework -- the ME method -- can handle arbitrary priors and arbitrary constraints. It includes MaxEnt and Bayes' rule as special cases and, therefore, it unifies entropic and Bayesian methods into a single general inference scheme. The ME method goes beyond the mere selection of a single posterior, but also addresses the question of how much less probable other distributions might be, which provides a direct bridge to the theories of fluctuations and large deviations.