A finite-group hierarchical model has a conditional law of its latent tuple

Put $\mathrm{\Xi }=H\times {\mathrm{\Theta }}^m$ with its finite product sigma-algebra. The model definition supplies the proper latent prior $P_0$ on $\mathrm{\Xi }$ and the probability kernel from $\mathrm{\Xi }$ to the product observation space $\mathcal{Y}$; apply [F2] to obtain the joint probability law $P_0(d\xi )K(\xi ,dy)$ on $\mathrm{\Xi }\times \mathcal{Y}$. Let $\mathrm{\Omega }=\mathrm{\Xi }\times \mathcal{Y}$ with this law and let $\xi$ and $Y$ be the coordinate projections. By [F5], both projections are measurable.

The target $\mathrm{\Xi }$ is standard Borel by [F3], since $H$ and $\mathrm{\Theta }$ are standard Borel and there are finitely many factors. The coordinate map $\xi :\mathrm{\Omega }\to \mathrm{\Xi }$ is a measurable random element, and $\sigma (Y)$ is a sub-sigma-algebra of the canonical product sigma-algebra. By [F4] under [A1], $\xi$ has a regular conditional distribution given $\sigma (Y)$. The definition of a regular conditional distribution requires every section to be a probability measure, including at exceptional sample points. The theorem asserts no normalization formula on a fixed observation fiber.