# Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks +
## 5 Experiment
### Optimal Settings for HMP-GMR  
HMP-GMR requires initial settings (i.e., \(nc\) and \(it\)), which influence accuracy and execution time. To find good initial settings, the HMP-GMR-OS algorithm was introduced in Section 4. With this algorithm, we use the following experiment setting: (1) the Hybrid BNs (i.e., CLG BNs 1, 2, 3 and 4 with \(n\) = 7), (2) the maximum execution time (i.e., \(max\_time\) = 3000 ms), (3) the number of samples (i.e., \(num\_samples\) = 50), (4) the upper limit on the maximum allowable number of components (i.e., \(ul\_max\_nc\) = 10), (5) the upper limit on the maximum number of iterations (i.e., \(ul\_max\_it\) = 10) and (6) the Hybrid-JT algorithm to obtain correct inference results. Figure 8 shows the results from this experiment obtained by the HMP-GMR-OS algorithm. Results for CLG BNs 1, 2, 3 and 4 are shown in Figure 8a-d, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **CLG BN 1** & **CLG BN 2** & **CLG BN 3** & **CLG BN 4** \\ \hline Converged & avg. KL-divergence & 0.0001 (0.0001) & 1.0352 (1.4604) & 2.2469 (1.452) & 3.6409 (3.1155) \\ Converged & avg. Time & 1514 (363.53) & 16547 (24113) & 2163.7 (415.6) & 9584 (9739.4) \\ Diverged & avg. KL-divergence & - & 106.3524 (16.0374) & - & - \\ Diverged & avg. Time & - & 9677.7 (5703.3) & - & - \\ Out of time & avg. KL-divergence & 0.0528 (0.0545) & 3.0795 (2.3361) & - & 82.0464 (121.4148) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average accuracies and average execution times for the four CLG BNs. Figure 7: Accuracies for convergence on the four CLG BNs. Table 4 shows minimum averages for KL-divergences in Figure 8 and best values for \(nc\) and \(it\) (numbers in parentheses are standard deviations). For example, for CLG BN 1, the minimum average for KL-divergence was 0.78 and its standard deviation was 3.37 at \(nc=5\) and \(it=10\). For CLG BN 4, the minimum average for KL-divergence was 11.37 and its standard deviation was 8.44 at \(nc=1\) and \(it=8\). For CLG BNs 1 and 3, the optimal number of iterations was the upper limit of 10. A better value might have been found if a larger number of iterations had been investigated. For CLG BNs 2 and 4, the best value for the maximum number of iterations was smaller than 10. Note that better results might be obtained, if the number of samples was increased and/or the ranges of \(nc\) and/or \(it\) were expanded. From this experiment, we observed that good accuracy can be achieved with a small number of components. Although the best values found by our experiment for \(nc\) were 5, 4, 6 and 1 for CLG BNs 1, 2, 3 and 4, respectively, the accuracy was not much better than using a single component. For example, the average of KL-divergence for CLG BN 1 was 0.78 at \(nc=5\) and \(it=10\), while the average of KL-divergence for CLG BN 1 was 1.09 at \(nc=1\) and \(it=10\). To check whether the difference was statistically significant, paired t-tests were performed at the 5% significance level. Table 5 shows confidence intervals from the paired t-tests. From these tests, we observed that the difference between the setting found by HMP-GMR-OS and the case with \(nc=1\) and \(it=10\) was not statistically significant for any of the CLG BNs. \begin{table}
\begin{tabular}{c c c c c} \hline  & **CLG BN 1** & **CLG BN 2** & **CLG BN 3** & **CLG BN 4** \\ \hline Minimum average & 0.7771 (3.3748) & 15.1868 (20.999) & 1.9617 (1.72) & 11.3698 (8.4352) \\ Best \(nc\) & 5 & 4 & 6 & 1 \\ Best \(it\) & 10 & 8 & 10 & 8 \\ \hline \end{tabular}
\end{table} 
Table 4: Minimum averages for Kullback Leibler (KL)-divergences and best values for \(nc\) and \(it\). Figure 8: Accuracy as a function of maximum allowable numbers of components and maximum iterations for four CLG BNs. This result suggests choosing \(nc=1\) as a default setting for HMP-GMR. This default setting for HMP-GMR was evaluated for accuracy against the Likelihood Weighting (LW) algorithm. Figure 9 shows accuracy comparison between HMP-GMR with the default setting \(nc=1\) and LW sampling for the four CLG BNs.