A shared hyperparameter can couple group parameters

The finite discrete spaces are standard Borel by [F4]. The two-point hyperprior is proper, $G_0={\delta }_0$ and $G_1={\delta }_1$ are probability kernels, and each singleton observation measure ${\nu }_j$ is finite, hence sigma-finite. Its observation kernel has the specified density $f_j({\theta }_j,\ast )=1$, jointly measurable on the finite product space. Thus the hierarchy in [F1] is well-defined and its joint evidence is $Z(\ast )=1$. By [F2], the full posterior equals the prior: it assigns probability $1\mathrm{/}2$ to $(\eta ,{\theta }_1,{\theta }_2)=(0,0,0)$ and $1\mathrm{/}2$ to $(1,1,1)$.

For either fixed $\eta$, the conditional marginal likelihood is $m(\ast \mid \eta )=1$, so both values attain the maximum. The map from the singleton data space to $\eta =0$ is measurable, hence [F3] permits the plug-in choice $\widehat{\eta }(\ast )=0$; its posterior is ${\delta }_0\otimes {\delta }_0$.

Under the full posterior in step 1.1, $\mathrm{Pr}({\theta }_1=1)=\mathrm{Pr}({\theta }_2=1)=1\mathrm{/}2$ but $\mathrm{Pr}({\theta }_1=1,{\theta }_2=1)=1\mathrm{/}2\mathrm{\ne }1\mathrm{/}4$. Therefore the group parameters are dependent after the shared hyperparameter is integrated out, and the full posterior differs from the plug-in posterior in step 2.1.