Current-group and new-group posterior predictions are probability kernels

Fix a current group $j$ and $y\in \mathcal{Y}$. For each measurable future-event set $A$, the formula $Q_{j,y}^{\mathrm{c}\mathrm{u}\mathrm{r}}(A)=\int K_j({\theta }_j,A)P_y(d\xi )$ gives zero on $A=\mathrm{\varnothing }$ and one on the full observation space by [F1] and [F2]. If $(A_k{)}_{k\ge 0}$ are disjoint, then $K_j({\theta }_j,{\bigcup }_k{A}_k)={\lim }_N{\sum }_{k\le N}{K}_j({\theta }_j,A_k)$; [F4] passes this increasing limit through the $P_y$ integral and gives countable additivity. The integrand is measurable in $\xi$ by the kernel definition and coordinate projection, so [F3] makes $y\mapsto Q_{j,y}^{\mathrm{c}\mathrm{u}\mathrm{r}}(A)$ measurable against the posterior kernel $y\mapsto P_y$. Thus each $Q_j^{\mathrm{c}\mathrm{u}\mathrm{r}}$ is a probability kernel from $\mathcal{Y}$ to ${\mathcal{Y}}_j$.

For a new group, define $R_{\eta }(A)=\int K_{\mathrm{n}\mathrm{e}\mathrm{w}}(\theta ,A)G_{\eta }(d\theta )$. The empty-set and whole-space evaluations are zero and one. For disjoint $(A_k)$, countable additivity of each $K_{\mathrm{n}\mathrm{e}\mathrm{w}}(\theta ,\cdot )$ and [F4] pass the increasing partial sums through the $G_{\eta }$ integral; hence $R_{\eta }$ is a probability measure. [F3] makes $\eta \mapsto R_{\eta }(A)$ measurable, so $R$ is a probability kernel. Mixing $R_{\eta }(A)$ over $P_y$ through the measurable coordinate $\eta$ and repeating the countable-additivity argument shows that $Q_y^{\mathrm{n}\mathrm{e}\mathrm{w}}(A)$ is a probability measure and [F3] makes its evaluations measurable in $y$. Thus $Q^{\mathrm{n}\mathrm{e}\mathrm{w}}$ is a probability kernel from $\mathcal{Y}$ to $\tilde{\mathcal{Y}}$. Its integral through the marginal $P_y^{\eta }$ is the same iterated-kernel law.

For either prediction let $H(\xi ,A)$ be its conditional future-event probability given the latent tuple: $K_j({\theta }_j,A)$ for a current group, or $R_{\eta }(A)$ for a new group. Steps 1.1–1.2 make $H$ a probability kernel. In the stipulated extension, [F7] ensures that the product of the observation kernel and this future kernel is a probability kernel from $\mathrm{\Xi }$ to the product of data and future-observation spaces; applying [F6] with prior $P_0$ gives a joint probability law on $(\xi ,Y,\tilde{Y})$. Because every section of $H$ has total mass one, marginalizing out $\tilde{Y}$ leaves the original joint law of $(\xi ,Y)$, so [F1] still supplies its posterior conditional law $P_y$. For data event $B$ and future event $A$, the triple law gives $P(Y\in B,\tilde{Y}\in A)=E[1_{\{Y\in B\}}H(\xi ,A)]$. Applying the event identity in [F5] to $H(\xi ,A)$ turns the right side into ${\int }_B\int H(\xi ,A)P_y(d\xi )P_Y(dy)={\int }_B{Q}_y(A)P_Y(dy)$. This verifies the conditional-event identity for every data event $B$ and proves that both predictive kernels are the conditional laws of the future observations. Since $P_y=P_0$ off ${\mathcal{Y}}_{+}$, their exceptional-fiber predictions are the prior-predictive laws.