## 4 Application to Gaussian Graphical Models
### Junction Tree Representation
#### 4.3.2 Experiment 2: Normalizing Constants of General Graphs  
Next, we investigate the performance of the two estimators for more general graphs. The results of the previous experiment showed that both GNORM and EPSOM-HybJT are very close to the true normalizing constants. For general graphs that do not satisfy the conditions that previously allowed us to calculate the exact GW normalizing constant, we compare the values of the GNORM and EPSOM-HybJT estimates against each other and keep track of the runtime relative to that of the EPSOM-HybJT algorithm when \(p=10\). In Table 3, for \(p\leq 50\), very little separates the log normalizing constant estimates, and GNORM even demonstrates its strength in low dimensions with a runtime that is about 50 times faster than EPSOM-HybJT for \(p=10\). However, EPSOM-HybJT flips the script in all subsequent experiments and scales better as \(p\) grows. For \(p=60\), the GNORM estimator fails to give a finite estimate for the log normalizing constant. In these experiments, each graph is generated using Bernoulli draws to determine the existence of its edges and checked to ensure that the graph is non-decomposable; otherwise, this would simply reduce the problem to a summation of tractable IW constants and would be unable to fairly assess the approximating ability of EPSOM-HybJT. Contrast these results with the examples from the previous section where the scale matrix was assumed to be block-diagonal and the dependence structure was relatively simple. In those simpler \begin{table}
\begin{tabular}{l l r r r} \hline \hline
**Dimension** & **Method** & **Mean** & **MRE** & **Runtime** \\ \hline \multirow{2}{*}{\(p=5,d=12\)} & EPSOM-HybJT & 920.16 & \(6.46\times 10^{-4}\) & 1 \\  & GNORM & 920.17 & \(6.63\times 10^{-4}\) & 0.03 \\ \multirow{2}{*}{\(p=50,d=120\)} & EPSOM-HybJT & 9201.55 & \(6.44\times 10^{-4}\) & 9.60 \\  & GNORM & 9201.74 & \(6.63\times 10^{-4}\) & 35.94 \\ \multirow{2}{*}{\(p=100,d=240\)} & EPSOM-HybJT & 18,403.14 & \(6.45\times 10^{-4}\) & 19.26 \\  & GNORM & 18,403.46 & \(6.63\times 10^{-4}\) & 796.07 \\ \multirow{2}{*}{\(p=150,d=360\)} & EPSOM-HybJT & 27,604.72 & \(6.46\times 10^{-4}\) & 28.92 \\  & GNORM & 27,605.18 & \(6.63\times 10^{-4}\) & 4587.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean estimate, mean relative error (MRE), and relative runtime of EPSOM-HybJT and GNORM approximations to the log normalizing constant of the \(\text{GW}_{G_{p}}\big{(}\delta,\Lambda_{p}\big{)}\) density. \(G_{p}\) has \(p=5r\) vertices and \(d=12r\) free elements. Here, \(\delta=100\) and \(\Lambda_{p}=I_{r}\otimes\Lambda\). Estimates are reported over 20 replications, each using 1000 samples from the target distribution. The runtime of the GNORM algorithm is calculated relative to the runtime of the EPSOM-HybJT algorithm. \begin{table}
\begin{tabular}{l l r r r} \hline \hline
**\# Vertices** & **Method** & **Mean** & **SD** & **Runtime** \\ \hline \multirow{2}{*}{\(p=10\)} & EPSOM-HybJT & \(-2477.40\) & 0.003 & 1 \\  & GNORM & \(-2468.19\) & 0 & 0.02 \\ \multirow{2}{*}{\(p=30\)} & EPSOM-HybJT & \(-7450.69\) & 0.012 & 2.99 \\  & GNORM & \(-7428.00\) & 0.773 & 12.34 \\ \multirow{2}{*}{\(p=40\)} & EPSOM-HybJT & \(-10,030.95\) & 0.015 & 4.68 \\  & GNORM & \(-10,003.81\) & 1.596 & 42.76 \\ \multirow{2}{*}{\(p=50\)} & EPSOM-HybJT & \(-12,563.87\) & 0.014 & 6.76 \\  & GNORM & \(-12,528.42\) & 2.385 & 108.87 \\ \multirow{2}{*}{\(p=60\)} & EPSOM-HybJT & \(-15,170.05\) & 0.017 & 15.26 \\  & GNORM & \(-\)Inf & — & 231.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean estimate, standard deviation, and relative runtime of EPSOM-HybJT and GNORM approximations to the log normalizing constant of the \(\text{GW}_{G_{p}}(\delta,\Lambda)\). Here, \(\Lambda\) is not block-diagonal. Estimates are reported over 20 replications, each using 1000 samples from the corresponding G-Wishart distribution. The runtime of the GNORM algorithm is calculated relative to the runtime of the EPSOM-HybJT algorithm.