# Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks +
## 5 Experiment
### Optimal Settings for HMP-GMR
We use the following experiment setting: (1) type of the hybrid BNs (i.e., CLG BNs 1, 2, 3 and 4 with \(n=7\)), (2) type of inference algorithm (i.e., HMP-GMR at \(nc=1\) and \(it=10\) and LW sampling), (3) 100 samples, (4) the maximum execution time (i.e., \(max\_time=3000\) ms) and (5) the Hybrid-JT algorithm to obtain correct inference results. Figure 9 shows results from this experiment. When HMP-GMR didn't converge, it stopped at the maximum execution time and provided its current solution. The chart contains eight lanes for four groups (CLG BNs 1, 2, 3 and 4). In the two adjoined lanes for each group, the left lane denotes the HMP-GMR case, while the right lane denotes the LW case. For example, the first lane in Figure 9 denotes the HMP-GMR case for CLG BN 1 (i.e., HG 1), while the second lane in Figure 9 denotes the LW case for CLG BN 1 (i.e., LW 1). The execution times for the two cases in each group were set to similar values. That is, the number of samples for LW was controlled to achieve similar execution times as HMP-GMR. Table 6 shows averages of KL-divergences for the two algorithms (numbers in parentheses are standard deviations). For example, for CLG BN 1, an average KL-divergence from HMP-GMR was 0.57. For CLG BN 4, an average KL-divergence from LW was 14.29. The fourth row denotes a natural-log ratio between HMP-GMR and LW. In the comparison between HMP-GMR and LW, LW was better than HMP-GMR for CLG BN 2. \begin{table}
\begin{tabular}{c c c c c} \hline  & **CLG BN 1** & **CLG BN 2** & **CLG BN 3** & **CLG BN 4** \\ \hline HMP-GMR & 0.5665 (2.4692) & 12.224 (18.0106) & 2.6175 (1.6649) & 11.6846 (10.8626) \\ LW & 7.8211 (4.127) & 6.1218 (5.0357) & 4.7952 (6.865) & 14.2882 (24.955) \\ LN(HMP-GMR / LW) & \(-\)2.6251 & 0.6915 & \(-\)0.6054 & \(-\)0.2012 \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison between three algorithms on averages of KL-divergences. \begin{table}
\begin{tabular}{l c c c c} \hline  & **CLG BN 1** & **CLG BN 2** & **CLG BN 3** & **CLG BN 4** \\ \hline Confidence Interval & \(-\)1.7623, 1.1444 & \(-\)12.7994, 5.5235 & \(-\)1.1426, 0.1924 & \(-\)4.4534, 2.7875 \\ \hline \end{tabular}
\end{table} 
Table 5: Confidence intervals from paired t-tests between the optimal settings and the default settings with \(nc=1\) and \(it=10\). Figure 9: KL-divergences of HMP-GMR (HG) and likelihood weighting (LW) for four CLG BNs. Figure 10 shows accuracy comparison between LW and HMP-GMR for the four CLG BNs. For CLG BN 1, HMP-GMR provided much better accuracy than LW. For CLG BN 2, LW provided better accuracy than HMP-GMR. For CLG BN 3, HMP-GMR provided better accuracy than LW. For CLG BN 4, LW and HMP-GMR performed similarly but as can be seen in Figure 9, the results for LW were more variable. Accuracy from HMP-GMR was lower in comparison with LW, for the CLG BN containing many loops. CLG BN 2 contained 501 cycles, while CLG BNs 1 and 4 contained 21 cycles and CLG BN 3 contained no cycles. More extensive investigations would be needed to determine whether this superiority of LW over HMP-GMR generalizes to arbitrary BNs with many loops. Accuracies from HMP-GMR did not depend much on the number of discrete states. CLG BN 1 contained four discrete states, while the CLG BN 3 contained 16384 configurations of the discrete states. For these networks, HMP-GMR provided better accuracy than LW regardless of how many discrete states a CLG BN contains. We can consider whether to use HMP-GMR or LW according to the features of the CLG BN. The following list shows suggestions in which HMP-GMR can be chosen or not in terms of the number of configurations of the discrete states and the number of cycles. 1. **Small number of configurations of the discrete states and small number of cycles**: In this case, our experiment showed better accuracy from HMP-GMR in comparison with LW under a given time restriction. LW requires many samples to improve accuracy, while HMP-GMR uses message passing approach, which can provide exact results for a polytree network (1).