Generalized Bayes loss update and temperature

Let $(\mathrm{\Theta },\mathcal{T})$ be a standard Borel parameter space (Standard Borel spaces). Let $\mathsf{X}$ be a set of possible observations, and let $\mathrm{\ell }:\mathrm{\Theta }\times \mathsf{X}\to \mathbb{R}$ be a loss whose section ${\mathrm{\ell }}_x(\theta ):=\mathrm{\ell }(\theta ,x)$ is $\mathcal{T}$-measurable for every $x\in \mathsf{X}$. A prior is a probability measure $\pi$ on $(\mathrm{\Theta },\mathcal{T})$ (Probability measures and probability spaces). A temperature is a fixed scalar $w>0$ that multiplies the observed loss.

For $x\in \mathsf{X}$, define the normalizer $Z_w(\pi ,x):={\int }_{\mathrm{\Theta }}\exp \{-w{\mathrm{\ell }}_x(\theta )\}\pi (d\theta )\mathrm{.}$ The integrand is measurable by continuity of the exponential and composition of measurable maps (The exponential function is strictly increasing, Composition with a Borel measurable outer map preserves measurability). When $0<Z_w(\pi ,x)<\mathrm{\infty }$, the generalized Bayes loss update (or Gibbs loss update) is the set function $G_{w,\pi ,x}(A):=\frac{1}{Z_w(\pi ,x)}{\int }_A\exp \{-w{\mathrm{\ell }}_x(\theta )\}\pi (d\theta ),A\in \mathcal{T}\mathrm{.}$ It is a positive scalar multiple of the measure with density $\exp \{-w{\mathrm{\ell }}_x\}$ relative to $\pi$ (The measure with density $f$ relative to $\mu$). If the normalizer is zero or infinite, this definition supplies no update.

• This is a fixed-observation update from a supplied probability prior. It does not specify a sampling model or claim that the loss is a likelihood.
• The standard Borel assumption is the scope adopted by the BCS-19 design; the fixed-fiber normalization formula itself uses only the stated measurable space and probability prior.