Bayesian models, priors, and posterior laws

Let $\mathsf{E}=((\mathsf{Y},\mathcal{Y}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment (Statistical experiments). Equip the parameter set with a specified measurable structure $(\mathrm{\Theta },\mathcal{T})$ (Measurable spaces and measurable sets). A proper Bayesian model based on this experiment consists of a prior probability measure $\mathrm{\Pi }$ on $(\mathrm{\Theta },\mathcal{T})$ (Probability measures and probability spaces) and a probability kernel $K$ from $(\mathrm{\Theta },\mathcal{T})$ to $(\mathsf{Y},\mathcal{Y})$ (Measure kernel and probability kernel) that represents the experiment: $K(\theta ,B)=P_{\theta }(B)(\theta \in \mathrm{\Theta },B\in \mathcal{Y})\mathrm{.}$ Thus $\theta \mapsto K(\theta ,B)$ is $\mathcal{T}$-measurable for every data event $B$, as required for a parameter-measurable model (Parameter-measurable statistical models as kernels). The family of measures in a parametric model alone does not supply this parameter sigma-algebra or measurability condition (Parametric statistical models).

The word proper means that the prior is a probability measure. A sigma-finite improper prior is not a proper Bayesian model in this definition; fixed-data posterior normalizers for such priors are handled separately.

Let a proper Bayesian model be given, and let $J$ be its joint prior-predictive probability on $(\mathrm{\Theta }\times \mathsf{Y},\mathcal{T}\otimes \mathcal{Y})$, supplied by A prior and model kernel define a joint law. More generally, let $(\mathrm{\Omega },\mathcal{F},Q)$ be any probability space carrying parameter and data random elements $X:\mathrm{\Omega }\to \mathrm{\Theta }$ and $Y:\mathrm{\Omega }\to \mathsf{Y}$ whose joint law is $J$ (Probability measures and probability spaces, Random elements and real random variables, Law or distribution of a random element). A posterior regular conditional distribution (posterior RCD) is a regular conditional distribution of $X$ given $\sigma (Y)$, in the sense of Regular conditional distribution. It is a kernel indexed by the sample point $\omega \in \mathrm{\Omega }$. The canonical realization is $\mathrm{\Omega }=\mathrm{\Theta }\times \mathsf{Y}$ with $Q=J$ and coordinate maps $X(\theta ,y)=\theta$, $Y(\theta ,y)=y$.

A data-indexed posterior kernel is a probability kernel ${\mathrm{\Pi }}_{\cdot }$ from $(\mathsf{Y},\mathcal{Y})$ to $(\mathrm{\Theta },\mathcal{T})$ (Measure kernel and probability kernel) for which $\omega \mapsto {\mathrm{\Pi }}_{Y(\omega )}$ is a conditional law of $X$ given $Y$ (Conditional law given a random element). Equivalently, for every $A\in \mathcal{T}$ and $B\in \mathcal{Y}$, $Q(X\in A,Y\in B)={\int }_B{\mathrm{\Pi }}_y(A)Q_Y(dy),$ where $Q_Y$ is the data marginal (equal to $J_Y$). This kernel is indexed by data values, not sample points.

The definition asserts no existence of either conditional kernel. Values of a data-indexed version on a measurable $J_Y$-null set are not fixed by the conditional-law identities; replacing them there by a fixed probability preserves those identities. A dominated-density construction is a separate route: Bayes formula for dominated kernels requires a proper prior, a sigma-finite data measure, and a specified jointly measurable sampling density, and fills fibres whose marginal density is zero or infinite with the prior. It does not require the data space to be standard Borel.

Assume AC (The Axiom of Choice). Let a proper Bayesian model have joint prior-predictive law $J$, and let $(\mathrm{\Omega },\mathcal{F},Q)$ be any probability space carrying parameter and data random elements $X,Y$ whose joint law is $J$ (Random elements and real random variables, Law or distribution of a random element). If $(\mathrm{\Theta },\mathcal{T})$ is standard Borel (Standard Borel spaces), then a posterior RCD of $X$ given $\sigma (Y)$ exists. The data space $(\mathsf{Y},\mathcal{Y})$ need only be measurable. In particular, the claim applies to the canonical product realization supplied by A prior and model kernel define a joint law.

Assume AC (The Axiom of Choice). In a proper Bayesian model, suppose both parameter and data spaces $(\mathrm{\Theta },\mathcal{T})$ and $(\mathsf{Y},\mathcal{Y})$ are standard Borel (Standard Borel spaces). Let $J$ be the joint prior-predictive law and let $J_Y$ be its data marginal. There is a data-indexed posterior probability kernel $y\mapsto {\mathrm{\Pi }}_y$ such that, for all $A\in \mathcal{T}$ and $B\in \mathcal{Y}$, $J(A\times B)={\int }_B{\mathrm{\Pi }}_y(A)J_Y(dy),J_Y(B)={\int }_{\mathrm{\Theta }}K(\theta ,B)\mathrm{\Pi }(d\theta )\mathrm{.}$ The composition ${\mathrm{\Pi }}_{Y(\omega )}$ is a posterior RCD. Any two data-indexed posterior kernels agree as probability measures outside one measurable $J_Y$-null set. On any measurable $J_Y$-null set, the conditional-law identities do not select a pointwise value: replacing the kernel there by the prior $\mathrm{\Pi }$ preserves the identities.