Latent-variable joint and observed-data factorization

Let $(\mathrm{\Theta },\mathcal{T})$, $(\mathsf{X},\mathcal{A})$, and $(\mathsf{Z},\mathcal{C})$ be measurable spaces (sets equipped with sigma-algebras; Measurable spaces and measurable sets). Let $\nu$ and $\mu$ be sigma-finite reference measures on the observation and latent spaces; each has a countable measurable cover by sets of finite measure (Finite, sigma-finite, and semifinite measures). The finite product sigma-algebra is generated by measurable rectangles (The product sigma-algebra and its finite iterates). The product reference measure $\nu \otimes \mu$ is the one supplied by For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique.

A jointly measurable latent-variable density is a specified function $p:\mathrm{\Theta }\times \mathsf{X}\times \mathsf{Z}⟶[0,\mathrm{\infty }]$ measurable for $\mathcal{T}\otimes \mathcal{A}\otimes \mathcal{C}$ and normalized on every parameter fibre: ${\int }_{\mathsf{X}\times \mathsf{Z}}p(\theta ,x,z)(\nu \otimes \mu )(d(x,z))=1(\theta \in \mathrm{\Theta })\mathrm{.}$ Here $x$ is the observed coordinate and $z$ the latent coordinate. The observed marginal density is $m(\theta ,x):={\int }_{\mathsf{Z}}p(\theta ,x,z)\mu (dz)\mathrm{.}$ The valid-fibre set is $V:=\{(\theta ,x):0<m(\theta ,x)<\mathrm{\infty }\}$. Only on $V$ define the normalized latent-fibre density $q(z\mid \theta ,x):=\frac{p(\theta ,x,z)}{m(\theta ,x)}\mathrm{.}$ This definition requires a single jointly measurable density as an input. Likelihood densities supplies neither a sigma-algebra on $\mathrm{\Theta }$ nor joint parameter measurability from separately specified likelihood sections; the joint measurability above is an additional hypothesis. Because $p$ is already supplied as one joint function, this definition uses no choice axiom. It does not assert that AC can turn separately specified pointwise likelihood versions into a jointly measurable version. No quotient or pointwise conditional density is asserted on a zero or infinite marginal fibre.