When Is Generalized Bayes Bayesian? A Decision-Theoretic Characterization of Loss-Based Updating
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912868150018048 |
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| author | McAlinn, Kenichiro Takanashi, Kōsaku |
| author_facet | McAlinn, Kenichiro Takanashi, Kōsaku |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01573 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | When Is Generalized Bayes Bayesian? A Decision-Theoretic Characterization of Loss-Based Updating McAlinn, Kenichiro Takanashi, Kōsaku Methodology Machine Learning 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. |
| title | When Is Generalized Bayes Bayesian? A Decision-Theoretic Characterization of Loss-Based Updating |
| topic | Methodology Machine Learning |
| url | https://arxiv.org/abs/2602.01573 |