 density of $\Th$ given $\Om$ and $W$. To make it easy to derive sufficient conditions that $\ph_H$ is minimax, we show the following lemma.

\[lem:alter\_m(W)\] The marginal density $m(W)$ can alternatively be represented as $$m(W)=\int_{\Rc_r}f_\pi(\La;W)\dd\La,$$ where $$f_\pi(\La;W)=(2\pi v)^{-qr/2}|\La|^{q/2}\pi_2^J(\La)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]$$ with $$\pi_2^J(\La)
=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La|^{-r-1}\pi_2[\La\{v_1I_r+(1-v_1)\La\}^{-1}].$$

[**Proof.**]{}   Let $$\La(I_r-\La)^{-1}=v_1\Om(I_r-\Om)^{-1},\quad v_1=\frac{v}{v_0},$$ where $0_{r\times r}\prec\La\prec I_r$. Since $v^{-1}(I_r-\La)^{-1}=v^{-1}I_r+v_0^{-1}\Om(I_r-\Om)^{-1}$, we observe that $$\begin{aligned}
&\frac{1}{v}\Vert W-\Th\Vert^2+\frac{1}{v_0}\tr\{\Om(I_r-\Om)^{-1}\Th\Th^\top\}\\
&=\frac{1}{v}\tr\Big[(I_r-\La)^{-1}\{\Th-(I_r-\La)W\}\{\Th-(I_r-\La)W\}^\top\Big]
 +\frac{1}{v}\tr(\La WW^\top),\end{aligned}$$ so $\pi(\Th|\Om,W)$ is proportional to $$\pi(\Th|\Om,W)
\propto \exp\Big[-\frac{1}{2v}\tr\Big[(I_r-\La)^{-1}\{\Th-(I_r-\La)W\}\{\Th-(I_r-\La)W\}^\top\Big]\Big],$$ namely, $\Th|\Om,W\sim\Nc_{r\times q}((I_r-\La)W,v(I_r-\La)\otimes I_q)$. Integrating out (\[eqn:m(W)\]) with respect to $\Th$ gives that $$\label{eqn:m(W)-1}
m(W)=(2\pi v)^{-qr/2}\int_{\Rc_r} |\La|^{q/2}\pi_2(\Om)\exp\Big[-\frac{1}{2v}\tr(\La WW^\top)\Big]\dd\Om.$$ Note that $\Om=\La\{v_1I_r+(1-v_1)\La\}^{-1}$ and the Jacobian of the transformation from $\Om$ to $\La$ is given by $$J[\Om\to \La]=v_1^{r(r+1)/2}|v_1I_r+(1-v_1)\La)|^{-r-1}.$$ Hence making the transformation from $\Om$ to $\La$ for (\[eqn:m(W)-1\]) completes the proof.

Let $\Dc_\La$ be an $r\times r$ symmetric matrix of differentiation operators with respect to $\La=(\la_{ij})$, where the $(i,j)$-th element of $\Dc_\La$ is $$\{\Dc_\La\}_{ij}=\frac{1+\de_{ij}}{2}\frac{\partial}{\partial\la_{ij}}.$$ Proposition \[prp:cond\_mini\] and Lemma \[lem:alter\_m(W)\] are utilized to get sufficient conditions for minimaxity of $\ph_H$.

\[thm:faith\] Let $f_\pi(\La;W)$ and $\pi_2^J(\La)$ be defined as