Calibrating generalized Bayes temperature

In the objective of The generalized Bayes loss update uniquely minimizes loss plus KL, $w$ scales the data-loss term relative to the KL penalty. Replacing ${\mathrm{\ell }}_x$ by $c{\mathrm{\ell }}_x$ for $c>0$ has the same algebraic effect as replacing $w$ by $cw$; the numerical temperature therefore depends on the loss units.

Bissiri, Holmes, and Walker (2016, §§3.1–3.4) describe several model- or elicitation-specific ways to choose $w$: cross-validation and subjective selection; matching expected loss-to-data and loss-to-prior using a specified joint reference law (with an empirical alternative); putting a hierarchical loss prior on $w$; and matching posterior probability of a loss-based confidence interval to its nominal level or eliciting a posterior-to-prior odds change at reference points. These are calibration proposals with additional inputs, not consequences of the update formula.

An M-loss can be chosen because its risk minimizer is the target of interest. That target choice alone proves neither contamination robustness nor frequentist coverage. The finite counterexample on the companion page shows why a positive temperature cannot supply a general coverage guarantee.