Xi'an's Og

I spotted this title in the new arXiv postings on Monday. When Is Generalized Bayes Bayesian? A Decision-Theoretic Characterization of Loss-Based Updating by Kenichiro McAlinn  & Kōsaku Takanashi is discussing decision-theoretic consequences of generalized Bayes approaches based on losses and show that decisions based on a loss-based posterior coincides with those of ordinary Bayes if and only if the loss is essentially a negative log-likelihood (leading to a belief posterior). This is not very surprising in that, otherwise, there is no Bayesian update delivering the generalised Bayes pseudo-posteriors (which can be traced back to a 2007 result of Catoni). The authors also demonstrate that generalized marginal likelihoods are not delivering evidence for decision posteriors, and thus that Bayes factors are not well-defined in this context, which reminds me of our warning for ABC model choice. However, the reason here is much more mundane, as it is due to the decision posterior failing to identify the normalising constant Z(x). Outside belief posteriors. The paper concludes with a coherence book, which is a table reproduced above.