Type-II maximum marginal likelihood and empirical-Bayes plug-in posterior

Assume a finite-group hierarchy as in Finite-group hierarchical Bayesian model, with hyperparameter $\eta \in H$, group parameters ${\theta }_1,\dots ,{\theta }_m$, conditional probability kernel $G_{\eta }$ (Measure kernel and probability kernel) on the group-parameter space, and specified jointly measurable likelihood density $L_y({\theta }_{1:m})={\prod }_{j=1}^m{f}_j({\theta }_j,y_j)$ against the common product data measure. For fixed $\eta$, define its marginal likelihood $m(y\mid \eta ):={\int }_{{\mathrm{\Theta }}^m}{L}_y({\theta }_{1:m})G_{\eta }^{\otimes m}(d{\theta }_{1:m})\in [0,\mathrm{\infty }]\mathrm{.}$ Where the function $\eta \mapsto m(y\mid \eta )$ is finite-valued and attains a positive finite maximum, a Type-II maximum marginal likelihood choice is any $\widehat{\eta }(y)\in {\mathrm{arg\hspace{0.17em}max}}_{\eta \in H}m(y\mid \eta )\mathrm{.}$ This definition does not assert that a maximum exists. A data-indexed choice requires a selected measurable maximizer on its domain.

Only when $0<m(y\mid \widehat{\eta }(y))<\mathrm{\infty }$ define the corresponding empirical-Bayes plug-in posterior on a measurable $A\subseteq {\mathrm{\Theta }}^m$ by ${\mathrm{\Pi }}_{\widehat{\eta }(y),y}(A):=\frac{{\int }_A{L}_y({\theta }_{1:m})G_{\widehat{\eta }(y)}^{\otimes m}(d{\theta }_{1:m})}{m(y\mid \widehat{\eta }(y))}\mathrm{.}$ The finite-positive condition makes this a probability update by Fiberwise posterior propriety is equivalent to finite positive evidence.

Under a proper hyperprior $H_0$, at a data fiber with finite positive joint evidence $0<Z(y):={\int }_{H}m(y\mid \eta )H_0(d\eta )<\mathrm{\infty }$, full Bayes instead retains the posterior law of $\eta$ jointly with the group parameters, as in Hierarchical posterior normalization on positive finite evidence. Integrating over a shared uncertain hyperparameter can induce dependence between group parameters, while a plug-in update fixes that hyperparameter at one selected value; special cases can leave the groups independent. This definition asserts no generic empirical-Bayes risk, consistency, dominance, or credible-coverage guarantee.