Finite-group hierarchical Bayesian model

Fix a finite group count $m\ge 1$, a standard Borel hyperparameter space $(H,\mathcal{H})$, and a standard Borel parameter space $(\mathrm{\Theta },\mathcal{T})$. Let $\mathrm{\Pi }$ be a probability measure on $(H,\mathcal{H})$ and let $G:H\times \mathcal{T}\to [0,1]$ be a probability kernel. For each $j\in \{1,\dots ,m\}$, let $({\mathcal{Y}}_j,{\mathcal{A}}_j)$ be a measurable observation space, let ${\nu }_j$ be a sigma-finite measure on it, and let $K_j$ be a probability kernel from $(\mathrm{\Theta },\mathcal{T})$ to $({\mathcal{Y}}_j,{\mathcal{A}}_j)$ with a specified jointly measurable density $f_j:\mathrm{\Theta }\times {\mathcal{Y}}_j\to [0,\mathrm{\infty }]$: $K_j(\theta ,B)={\int }_B{f}_j(\theta ,y_j){\nu }_j(d{y}_j)(B\in {\mathcal{A}}_j)\mathrm{.}$

The finite-group hierarchical model is the following specified kernel construction. First draw $\eta \sim \mathrm{\Pi }$. Conditional on $\eta$, draw ${\theta }_1,\dots ,{\theta }_m$ from the finite product kernel $G_{\eta }^{\otimes m}(d{\theta }_1\cdots d{\theta }_m)={⨂}_{j=1}^m{G}_{\eta }(d{\theta }_j)\mathrm{.}$ Thus the group parameters are conditionally independent and have the same conditional law $G_{\eta }$. Conditional on the parameter tuple, draw the group observations from the product kernel ${⨂}_{j=1}^m{K}_j({\theta }_j,\cdot )$. The product-kernel laws are probability kernels by Finite products of probability kernels are probability kernels. For each fixed $\eta$, the coordinate maps under $G_{\eta }^{\otimes m}$ are independent random elements, so their joint law is the finite product law (Independent random elements have product joint law).

Put $\mathrm{\Xi }=H\times {\mathrm{\Theta }}^m$ with its finite product sigma-algebra. The proper latent prior $P_0$ on $\mathrm{\Xi }$ is $P_0(d\eta ,d{\theta }_1\cdots d{\theta }_m)=\mathrm{\Pi }(d\eta ){\prod }_{j=1}^m{G}_{\eta }(d{\theta }_j),$ well-defined by A prior and model kernel define a joint law. For $\xi =(\eta ,{\theta }_1,\dots ,{\theta }_m)\in \mathrm{\Xi }$, lift each observation kernel to the common source $\mathrm{\Xi }$ by $M_j(\xi ,B)=K_j({\theta }_j,B)(B\in {\mathcal{A}}_j)\mathrm{.}$ The coordinate projection $\xi \mapsto {\theta }_j$ is measurable, so the kernel definition shows that each $M_j$ is a probability kernel. Its specified jointly measurable density is $(\xi ,y_j)\mapsto f_j({\theta }_j,y_j)$, by composition with the same measurable coordinate projection. Let $\mathcal{Y}={\prod }_{j=1}^m{\mathcal{Y}}_j$, $\nu ={⨂}_{j=1}^m{\nu }_j$, and $y=(y_1,\dots ,y_m)$. The observation kernel from $\mathrm{\Xi }$ has jointly measurable density $L_y(\eta ,{\theta }_1\cdots {\theta }_m)={\prod }_{j=1}^m{f}_j({\theta }_j,y_j)$ with respect to $\nu$, by Finite products of dominated kernels have the product density. Its evidence is the possibly extended nonnegative number $Z(y)={\int }_{\mathrm{\Xi }}{L}_y(\eta ,{\theta }_1\cdots {\theta }_m)P_0(d\eta ,d{\theta }_1\cdots d{\theta }_m)\mathrm{.}$ The normalized likelihood measure is called a pointwise joint posterior on a fiber only when $0<Z(y)<\mathrm{\infty }$; no quotient is formed at zero or infinite evidence.

For each group, $\theta \mapsto K_j(\theta ,\cdot )$ is a parametric statistical model in the sense of Parametric statistical models, with its parameter-measurable kernel representation as in Parameter-measurable statistical models as kernels. Here “conditionally independent given $\eta$” is shorthand for the explicit finite product-kernel specification above. The library's random-element conditional-independence terminology is defined in Conditional independence given a sigma-algebra; its conditional-expectation-class convention assumes AC. The finite product-kernel definition here does not use that class convention or any choice axiom.

The product-kernel assumption is part of this model. An arbitrary finitely exchangeable law need not be a mixture of iid laws; no finite de Finetti theorem is claimed here. When $\eta$ is unknown, the joint posterior retains its uncertainty and induces dependence between group parameters after $\eta$ is integrated out.