Current-group and new-group posterior prediction

Let $\mathcal{Y}={\prod }_{j=1}^m{\mathcal{Y}}_j$ be the data space and ${\mathcal{Y}}_{+}=\{y\in \mathcal{Y}:0<Z(y)<\mathrm{\infty }\}$ its measurable subset of positive finite evidence fibers. Write $P_y$ for the full latent probability kernel supplied by Hierarchical posterior normalization on positive finite evidence, and set ${\mathrm{\Lambda }}_y:=P_y$ for every $y\in \mathcal{Y}$. On ${\mathcal{Y}}_{+}$ this is the normalized joint posterior; off ${\mathcal{Y}}_{+}$ it is the prior filler $P_0$, so no likelihood quotient is formed there. For an already observed group $j$, its current-group posterior predictive law for a future observation with the same conditional observation kernel is $Q_{j,y}^{\mathrm{c}\mathrm{u}\mathrm{r}}(A)={\int }_{\mathrm{\Xi }}{K}_j({\theta }_j,A){\mathrm{\Lambda }}_y(d\eta ,d{\theta }_1\cdots d{\theta }_m)\mathrm{.}$

For a new group, specify a probability kernel $K_{\mathrm{n}\mathrm{e}\mathrm{w}}$ from $(\mathrm{\Theta },\mathcal{T})$ to a future observation space $(\tilde{\mathcal{Y}},\tilde{\mathcal{A}})$. Its new-group posterior predictive law is $Q_y^{\mathrm{n}\mathrm{e}\mathrm{w}}(A)={\int }_{\mathrm{\Xi }}[{\int }_{\mathrm{\Theta }}{K}_{\mathrm{n}\mathrm{e}\mathrm{w}}(\theta ,A)G_{\eta }(d\theta )]{\mathrm{\Lambda }}_y(d\eta ,d{\theta }_1\cdots d{\theta }_m)\mathrm{.}$ For $y\in {\mathcal{Y}}_{+}$, if ${\mathrm{\Lambda }}_y^{\eta }$ denotes the hyperparameter marginal of ${\mathrm{\Lambda }}_y$, the joint posterior law of $(\eta ,{\theta }_{\mathrm{n}\mathrm{e}\mathrm{w}})$ used here is ${\mathrm{\Lambda }}_y^{\eta }(d\eta )G_{\eta }(d{\theta }_{\mathrm{n}\mathrm{e}\mathrm{w}})$. Thus the new-group prediction first draws a new latent parameter from $G_{\eta }$ and then draws its observation from $K_{\mathrm{n}\mathrm{e}\mathrm{w}}$. On ${\mathcal{Y}}_{+}$ both formulas average over the joint posterior and retain uncertainty about unknown hyperparameters. On exceptional fibers they reduce to prior-predictive laws because ${\mathrm{\Lambda }}_y=P_0$.