Conditional normal shrinkage in a finite hierarchy

Fix a group $j$ and specialize [F5] to one observation: identify its parameter with ${\theta }_j$, set ${\mu }_0=\mu$, ${\tau }_0^2={\tau }^2$, $n=1$, $y_1={\bar{y}}_j$, and ${\sigma }^2=s_j^2$. The model and pointwise density specification in [F1] supply this normal prior and the canonical normal likelihood density used by [F5], while [A1] and [A2] verify the choice and positive-variance hypotheses of [F5]. The theorem gives a finite positive group evidence and posterior variance $V_j=(1\mathrm{/}{\tau }^2+1\mathrm{/}{s}_j^2{)}^{-1}$ and mean ${\widehat{\theta }}_j=V_j(\mu \mathrm{/}{\tau }^2+{\bar{y}}_j\mathrm{/}{s}_j^2)$, as stated.

For each group put ${\mu }_j(A)={\int }_A{f}_j(\theta ,{\bar{y}}_j)N(\mu ,{\tau }^2)(d\theta )$ and let $m_j={\mu }_j(\mathbb{R})>0$. By [F5] and step 1.1, ${\mu }_j\mathrm{/}{m}_j=Q_j=N({\widehat{\theta }}_j,V_j)$. Apply [F3] to the hierarchy with the proper point-mass hyperprior at the fixed $(\mu ,{\tau }^2)$. Its latent prior on ${\mathbb{R}}^m$ is the product of the group priors by [F1], and its likelihood density is ${\prod }_j{f}_j({\theta }_j,{\bar{y}}_j)$ by [F2]. Repeated Tonelli [F4] shows that the normalized posterior assigns each measurable rectangle ${\prod }_j{A}_j$ mass ${\prod }_j{\mu }_j(A_j)\mathrm{/}{\prod }_j{m}_j={\prod }_j{Q}_j(A_j)$, the rectangle mass of ${⨂}_j{Q}_j$ by [F9]. By [F8], rectangles form a generating pi-system. Both are probability measures agreeing on it; the constant exhaustion by the full space has common finite mass one, so [F6] makes them equal. This proves conditional independence and the displayed formula for every group.

Now suppose $(\mu ,{\tau }^2)$ is random with a proper hyperprior supported on $H$. For fixed observed summaries, put $M_j((\mu ,{\tau }^2),A)={\int }_A{f}_j(\theta ,{\bar{y}}_j)N(\mu ,{\tau }^2)(d\theta )$. The factor $f_j(\cdot ,{\bar{y}}_j)$ is measurable by the section fact [F8]; composing it with the parameter-coordinate projection shows that $(\eta ,\theta )\mapsto 1_A(\theta )f_j(\theta ,{\bar{y}}_j)$ is product-measurable. Since $G_{\eta }=N(\mu ,{\tau }^2)$ is a probability kernel, [F7] makes $M_j(\cdot ,A)$ measurable. Hence $m_j=M_j(\cdot ,\mathbb{R})$ and $Q_j(A)=M_j(\cdot ,A)\mathrm{/}{m}_j$ are measurable, with $m_j>0$ by step 1.1. Repeated Tonelli [F4] and the full-posterior formula [F3] give, for measurable $C\subseteq H$ and $A_1,\dots ,A_m\subseteq \mathbb{R}$, $P_y(C\times {\prod }_j{A}_j)=Z(y{)}^{-1}{\int }_C{\prod }_j{M}_j((\mu ,{\tau }^2),A_j)\mathrm{\Pi }(d\mu ,d{\tau }^2)$. Taking every $A_j=\mathbb{R}$ shows ${\mathrm{\Pi }}_y(C)=Z(y{)}^{-1}{\int }_C{\prod }_j{m}_{j}d\mathrm{\Pi }$. Taking only $A_j=A$ and all other sets equal to $\mathbb{R}$ therefore gives $P_y({\theta }_j\in A)={\int }_H{Q}_j(A){\mathrm{\Pi }}_y(d\mu ,d{\tau }^2)$. By [F5] and step 1.1, $Q_j=N({\widehat{\theta }}_j,V_j)$, so the group posterior is its conditional normal law averaged over the full hyperparameter posterior. If ${\mathrm{\Pi }}_y$ is a point mass, this mixture equals its single conditional normal component; in general it need not equal a plug-in law.