Posterior-predictive probability kernel

Let $(\mathrm{\Theta },\mathcal{T})$ and $(\mathsf{Y},\mathcal{Y})$ be measurable spaces. Suppose $\mathrm{\Pi }$ is a supplied probability kernel from $\mathsf{Y}$ to $\mathrm{\Theta }$; write its probability measure at $y$ as ${\mathrm{\Pi }}_y$. Let $Q$ be a probability kernel from $\mathrm{\Theta }$ to a measurable future-observation space $(\mathsf{Z},\mathcal{Z})$. Define the posterior-predictive probability kernel by $Q_y^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}(B):={\int }_{\mathrm{\Theta }}{Q}_{\theta }(B){\mathrm{\Pi }}_y(d\theta ),y\in \mathsf{Y},B\in \mathcal{Z}\mathrm{.}$ A probability kernel has pointwise probability-measure sections and measurable event evaluations, as in Measure kernel and probability kernel. This definition takes the posterior kernel as supplied; it does not form a likelihood quotient.

For a proper-prior dominated model, Bayes formula for dominated kernels supplies one such data-indexed kernel when the sampling laws have a specified nonnegative jointly measurable density against a sigma-finite data measure. Its stated filler on zero or infinite normalizers is part of that kernel version.