Misfit evidence and the potential posterior law

Let $X$ be a separable real Banach space with its norm topology, $D\subseteq X$ a Borel forward-map domain, and ${\mu }_0$ a Borel probability measure on $X$ with ${\mu }_0(D)=1$. Write ${\mu }_{0,D}$ for its restriction to $D$, so ${\mu }_{0,D}(A)={\mu }_0(A)$ for $A\in \mathcal{B}(D):=\{D\cap B:B\in \mathcal{B}(X)\}$. Let $(\mathsf{Y},\mathcal{Y})$ be a measurable data space. A likelihood potential is a function $\mathrm{\Phi }:D\times \mathsf{Y}\to (-\mathrm{\infty },+\mathrm{\infty }]$ that is measurable from $(D\times \mathsf{Y},\mathcal{B}(D)\otimes \mathcal{Y})$ to $((-\mathrm{\infty },+\mathrm{\infty }],\mathcal{B}((-\mathrm{\infty },+\mathrm{\infty }]))$, where $\mathcal{B}((-\mathrm{\infty },+\mathrm{\infty }])$ is the trace of $\mathcal{B}(\bar{\mathbb{R}})$ on $(-\mathrm{\infty },+\mathrm{\infty }]$. Viewing $\mathrm{\Phi }$ via the inclusion $(-\mathrm{\infty },+\mathrm{\infty }]\hookrightarrow \bar{\mathbb{R}}$, it is also product-measurable as an extended-real-valued map because for every Borel set $B\subseteq \bar{\mathbb{R}}$, ${\mathrm{\Phi }}^{-1}(B)={\mathrm{\Phi }}^{-1}(B\cap (-\mathrm{\infty },+\mathrm{\infty }])$ is measurable by the definition of the trace sigma-algebra. Here $+\mathrm{\infty }$ is allowed and $-\mathrm{\infty }$ is excluded; set $e^{-\mathrm{\infty }}=0$. The scalar map $t\mapsto e^{-t}$ on this range is continuous at finite $t$, and is continuous at $+\mathrm{\infty }$: given $\epsilon >0$, the limit supplies $a\in \mathbb{R}$ such that $t>a$ implies $e^{-t}<\epsilon$, and the order-topology neighborhood $(a,+\mathrm{\infty }]$ maps into $[0,\epsilon )$. Hence it is Borel, so $L_{\mathrm{\Phi }}:D\times \mathsf{Y}\to [0,+\mathrm{\infty })$, $L_{\mathrm{\Phi }}(u;y):=e^{-\mathrm{\Phi }(u;y)}$, is product-measurable by composition, and each section $u\mapsto L_{\mathrm{\Phi }}(u;y)$ is nonnegative and measurable. Define its evidence by

$$Z_{\mathrm{\Phi }}(y):={\int }_D{L}_{\mathrm{\Phi }}(u;y){\mu }_{0,D}(du)\in [0,+\mathrm{\infty }]\mathrm{.}$$

For each fixed $y$ with $0<Z_{\mathrm{\Phi }}(y)<+\mathrm{\infty }$, define the normalized fiberwise update on $A\in \mathcal{B}(D)$ by ${\mathrm{\Pi }}^y(A):=\frac{{\int }_A{L}_{\mathrm{\Phi }}(u;y){\mu }_{0,D}(du)}{Z_{\mathrm{\Phi }}(y)}\mathrm{.}$ No normalized fiberwise update is defined by this formula when $Z_{\mathrm{\Phi }}(y)=0$ or $Z_{\mathrm{\Phi }}(y)=+\mathrm{\infty }$.

For a data-indexed claim, an observation model and its version convention must also be specified. In the dominated case, let $Q$ be a proper probability kernel from $(D,\mathcal{B}(D))$ to $(\mathsf{Y},\mathcal{Y})$, let $\nu$ be a sigma-finite measure on $(\mathsf{Y},\mathcal{Y})$, and suppose $Q(u,B)={\int }_B{L}_{\mathrm{\Phi }}(u;y)\nu (dy)$ for every $u\in D$ and $B\in \mathcal{Y}$. Then $Z_{\mathrm{\Phi }}$ is the density of the prior-predictive data law with respect to $\nu$; a data-indexed conditional-law version uses ${\mathrm{\Pi }}^y$ when $0<Z_{\mathrm{\Phi }}(y)<\mathrm{\infty }$ and the prior ${\mu }_{0,D}$ when $Z_{\mathrm{\Phi }}(y)$ is zero or infinite. These filled fibers have zero prior-predictive mass, and their chosen values are not normalized fiberwise updates. More generally, for a proper Bayesian model with standard-Borel parameter space $(D,\mathcal{B}(D))$ and data space $(\mathsf{Y},\mathcal{Y})$ (each measurably isomorphic to the Borel space of a Polish space), the Axiom of Choice guarantees a data-indexed posterior kernel; its values are unique as probability measures outside one data-marginal-null set, and values on that null set remain version choices. The dominated construction is choice-free; the separate standard-Borel existence branch assumes the Axiom of Choice.

A prior on the admissible domain is a Borel probability with full mass there, and its restriction is a probability.

A measurable space consists of a set and a sigma-algebra.

Joint measurability uses the product sigma-algebra.

The potential takes extended real values, allows positive infinity, and excludes negative infinity.

The extended real line carries its order-topology Borel sigma-algebra.

A function is measurable when preimages of measurable target sets are measurable in the source.

Every section of a product-measurable extended-real function is measurable.

Composing a measurable map with a Borel outer map preserves measurability.

The real exponential is continuous on finite inputs.

The exponential tends to zero at negative infinity, giving the likelihood map's value zero at positive infinity.

A continuous scalar map has Borel preimages of Borel sets.

A proper Bayesian model is a probability prior with a parameter-measurable probability sampling kernel.

A probability kernel has measurable evaluations and probability sections at every parameter.

The dominating data measure is sigma-finite.

Fixed-data evidence normalizers are defined for sigma-finite priors, but do not define a prior-predictive law for an improper prior.

A nonnegative likelihood update is a probability exactly when its evidence is positive and finite and the scale is the reciprocal.

A jointly measurable dominated sampling density gives the prior-predictive density and a conditional kernel with prior fillers on zero and infinite fibers.

A posterior RCD is sample-point-indexed; a data-indexed posterior kernel is a version and its null-set values are not fixed.

Changing a conditional kernel on a measurable data-marginal-null set preserves its conditional identities.

Standard Borel means measurably isomorphic to a Polish space with its Borel sigma-algebra.

The Axiom of Choice states that every family of nonempty sets has a choice function.

Under AC and standard-Borel parameter and data spaces, a data-indexed posterior kernel exists and is unique off one data-marginal-null set.

Dashti and Stuart, §3.2 and §4.1, full lecture notes

Latz, §2.2, Theorem 2.5 and its proof in Appendix B (printed p. 23), full paper