## Appendix C Hyper-Inverse Wishart Density  
We introduce some notation to help us obtain a closed form for the marginal likelihood in the case of a decomposable graph \(G\). For an \(n\times p\) matrix \(X\), \(X_{\mathbb{C}}\) is defined as the submatrix of \(X\) consisting of columns with indices in the clique \(C\). Let \((X_{1},X_{2},\ldots,X_{p})=(x_{1},x_{2},\ldots,x_{n})^{\prime}\), where \(X_{i}\) is the \(i\)th column of \(X_{n\times p}\). If \(C=\{i_{1},i_{2},\ldots,i_{|C|}\}\), where \(1\leq i_{1}<i_{2}<\ldots<i_{|C|}\leq p\), then \(X_{\mathbb{C}}=(X_{i_{1}},X_{i_{2}},\ldots,X_{i_{|C|}})\). For any square matrix \(A=(a_{ij})_{p\times p^{\prime}}\) define \(A_{\mathbb{C}}=(a_{ij})_{|C|\times|C|}\) where \(i,j\in C\), \(|C|\) is the cardinality of the clique \(C\), and the order of entries carries into the new sub-matrix \(A_{\mathbb{C}}\). Therefore, \(X_{\mathbb{C}}^{\prime}X_{\mathbb{C}}=(X^{\prime}X)_{\mathbb{C}}\).  
Decomposable graphs correspond to a special kind of sparsity pattern in \(\Sigma\), henceforth denoted \(\Sigma_{G}\). Suppose we have a \(\mathrm{H}\mathrm{i}\mathrm{i}\mathrm{p}_{G}(b,D)\) distribution on the cone of \(p\times p\) positive definite matrices with \(b>2\) degrees of freedom and a fixed \(p\times p\) positive definite matrix \(D\) such that the joint density factorizes on the junction tree of the given decomposable graph \(G\) as  
\[p(\Sigma_{G}\mid b,D)=\frac{\prod_{\mathbb{C}\in\mathcal{C}}p(\Sigma_{C}\mid b,D_{\mathbb{C}})}{\prod_{\mathbb{S}\in\mathcal{S}}p(\Sigma_{S}\mid b,D_{ \mathbb{S}})}, \tag{1}\]where for each \(C\in\mathcal{C},\Sigma_{C}\sim\mathrm{IW}_{C}(b,D_{C})\) has the density  
\[p(\Sigma_{C}\mid b,D_{C})\propto|\Sigma_{C}|^{-(b+2|C|)/2}\mathrm{etr}\Big{\{}- \frac{1}{2}\Sigma_{C}^{-1}D_{C}\Big{\}}, \tag{10}\]  
where \(\mathrm{etr}(\cdot)=\exp\big{\{}\mathrm{tr}(\cdot)\big{\}}\). Here, \(\mathrm{IW}(b,D)\) denotes the inverse-Wishart distribution with degrees of freedom \(b\) and a fixed \(p\times p\) positive definite matrix \(D\) with normalizing constant  
\[\bigg{|}\frac{1}{2}D\bigg{|}^{(b+p-1)/2}\Gamma_{d}^{-1}\bigg{(}\frac{b+p-1}{2} \bigg{)}.\]  
Note that we can establish equivalence to the parametrization used in Section 4.1 by taking \(\delta=b+p-1\). Since the joint density in Equation (9) factorizes over cliques and separators in the same way as in Equations (11) and (10),  
\[f(X\mid\Sigma_{G})=(2\pi)^{-\frac{up}{2}}\frac{\prod_{C\in\mathcal{C}}|\Sigma_ {C}|^{-\frac{up}{2}}\mathrm{etr}\Big{(}-\frac{1}{2}\Sigma_{C}^{-1}X_{C}^{ \prime}X_{C}\Big{)}}{\prod_{S\in\mathcal{S}}|\Sigma_{S}|^{-\frac{up}{2}} \mathrm{etr}\Big{(}-\frac{1}{2}\Sigma_{S}^{-1}X_{S}^{\prime}X_{S}\Big{)}}.\]  
Using the above equations, we can obtain the marginal likelihood,  
\[f(X\mid G)=(2\pi)^{-\frac{up}{2}}\frac{h(G,b,D)}{h(G,b+n,D+S)}=(2\pi)^{-\frac{up }{2}}\frac{\prod_{C\in\mathcal{C}}w(C)}{\prod_{S\in\mathcal{S}}w(S)}, \tag{11}\]  
where  
\[h(G,b,D)=\frac{\prod_{C\in\mathcal{C}}\Big{|}\frac{1}{2}D_{C}\Big{|}\frac{b+| C|-1}{2}\Gamma_{|C|}^{-1}\Big{(}\frac{b+|C|-1}{2}\Big{)}}{\prod_{S\in\mathcal{S}} \Big{|}\frac{1}{2}D_{S}\Big{|}^{\frac{b+|S|-1}{2}}\Gamma_{|S|}^{-1}\Big{(} \frac{b+|S|-1}{2}\Big{)}}, \tag{12}\]  
\[w(C)=\frac{|D_{C}|^{\frac{b+|C|-1}{2}}|D_{C}+X_{C}^{\prime}X_{C}|^{-\frac{b+n +|C|-1}{2}}}{2^{-\frac{n|C|}{2}}\Gamma_{|C|}\Big{(}\frac{b+|C|-1}{2}\Big{)} \Gamma_{|C|}^{-1}\Big{(}\frac{b+n+|C|-1}{2}\Big{)}}. \tag{13}\]