When Is Generalized Bayes Bayesian? A Decision-Theoretic Characterization of Loss-Based Updating

TL;DR

This paper provides a decision-theoretic framework distinguishing belief and decision posteriors, clarifies when loss-based posteriors align with Bayes, and characterizes generalized Bayes as an optimal decision rule under certain preferences.

Contribution

It offers a decision-theoretic characterization of loss-based updating, clarifies the conditions under which generalized Bayes coincides with traditional Bayes, and links generalized Bayes to entropy-penalized variational principles.

Findings

Loss-based posteriors equal Bayes only with negative log-likelihood loss.

Generalized marginal likelihood is not evidence for decision posteriors.

Generalized Bayes emerges as an optimal rule under entropy-penalized variational representation.

Abstract

Loss-based updating, including generalized Bayes, Gibbs, and quasi-posteriors, replaces likelihoods by a user-chosen loss and produces a posterior-like distribution via exponential tilt. We give a decision-theoretic characterization that separates \emph{belief posteriors} -- conditional beliefs justified by the foundations of Savage and Anscombe-Aumann under a joint probability mode l-- from \emph{decision posteriors} -- randomized decision rules justified by preferences over decision rules. We make explicit that a loss-based posterior coincides with ordinary Bayes if and only if the loss is, up to scale and a data-only term, negative log-likelihood. We then show that generalized marginal likelihood is not evidence for decision posteriors, and Bayes factors are not well-defined without additional structure. In the decision posterior regime, non-degenerate posteriors require nonlinear preferences over decision rules. Under sequential coherence and separability, these lead to an entropy-penalized variational representation yielding generalized Bayes as the optimal rule.