Maximum Entropy and Bayesian Conditioning Under Extended Space

TL;DR

This paper explores the relationship between Maximum Entropy and Bayesian conditioning when extending probability spaces to incorporate new information not initially represented as an event.

Contribution

It clarifies the conditions under which Bayesian conditioning in extended spaces aligns with Maximum Entropy, addressing criticisms and theoretical challenges.

Findings

Bayesian conditioning aligns with Maximum Entropy in extended spaces under certain conditions.

Critiques of Skyrms' approach are addressed, clarifying the role of space extension.

The paper discusses the limitations of Bayesian conditioning in non-degenerate models.

Abstract

This paper examines the conditions under which Bayesian conditioning aligns with Maximum Entropy. Specifically, I address cases in which newly learned information does not correspond to an event in the probability space defined on the sample space of outcomes. To facilitate Bayesian conditioning in such cases, one must therefore extend the probability space so that the new information becomes an event in this expanded space. Skyrms (1985) argues that Bayesian conditioning in an extended probability space on a product space of outcomes aligns precisely with the solution from Maximum Entropy. In contrast, Seidenfeld (1986) uses Friedman and Shimony's (1971) result to criticize Skyrms' approach as trivial, suggesting that alignment holds only under a degenerate probability model. Here, I argue that Friedman and Shimony's result must either (1) be a benign consequence of Skyrms' approach, or (2) pose a universal challenge to any method of extending spaces. Accepting (2) would imply that Bayesian conditioning is incapable of accommodating information beyond the probability space defined on the original outcome space.