# Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks +
## 5 Experiment
### Accuracy and Efficiency of HMP-GMR  
In this experiment, we investigated the accuracy and convergence of HMP-GMR for the four CLG BNs. To evaluate accuracy of HMP-GMR, exact inference results using Hybrid-JT inference were used. Some of the runs using Hybrid-JT stopped because of the exponential growth of components, so Hybrid-JT produced posterior distributions only for \(n\leq 7\). For this reason, this experiment used \(n=7\) for the four CLG BNs. Accuracy was measured by KL-divergence [38] (lower values  
Figure 4: Execution Times over \(n\) on CG BNs 1 and 2. Figure 5: Execution Times over \(n\) on CG BNs 3 and 4.  
mean better accuracy). We calculated the KL-divergence between exact and approximate results for each unobserved node and summed them (henceforth, we use KL-divergences to mean the sum of KL-divergences over unobserved nodes). The number of runs in the experiment was 100. The maximum allowable number of components was \(nc=2\). The maximum number of iterations was \(it=10\),000. The maximum execution time was \(max\_time=200\),000 millisecond (ms). In the experiment, there were three exit points: (1) When the algorithm converged, (2) when the time limit was exceeded and (3) when the algorithm diverged. When the algorithm did not converge, the algorithm halted and provided its current solution. Figure 6 shows percentages for each CLG BN for which the algorithm converged, diverged and ran out of time. For CLG BN 1, the algorithm converged in 97% of the runs and ran out of time in 3% of the cases (execution time > 200,000 ms). For CLG BN 2, the algorithm converged in 52% of the runs, ran out of time in 3% of the runs and diverged in 45% of the runs. For CLG BN 3, the algorithm converged in 100% of the runs. For CLG BN 4, the algorithm converged in only 31% of the runs and ran out of time in 69% of the runs. We observed that the large number of cycles (CLG BN 2) could cause many situations in which the algorithm did not converge (i.e., diverged or ran out of time). Also, the algorithm for the cases with no cycles (CLG BN 3) always converged. For the cases in which the algorithm ran out of time, if more time had been allowed it might have converged, diverged or failed to either converge or diverge (i.e., oscillated). In some cases for CLG BNs 1, 2 and 4, the algorithm oscillated until reaching the maximum execution time and halting. When there were many cycles and many discrete states (i.e., for CLG BN 4), the algorithm often ran out of time. Table 3 shows averages (avg.) for KL-divergence over runs and average execution times for three cases (converged, diverged and ran out of time) on the four CLG BNs (numbers in parentheses are standard deviations). Figure 7 shows the accuracy results from this experiment, when the algorithm converged. In Figure 7, the four lanes denote the four CLG BNs. For case of convergence, the averages for KL-divergence over runs of the experiment were 0.0001, 1.04, 2.25 and 3.64 for CLG BNs 1, 2, 3 and 4, respectively. Again, for the case of convergence, the average execution times were 1514, 16,547, 2164 and 9584 for CLG BNs 1, 2, 3 and 4, respectively. For the case of convergence, because of the large number (501) of cycles in CLG BN 2, the inference algorithm required more time (avg. 16,547 ms) to converge than others. Also, the algorithm for CLG BN 4 spent more time (avg. 9584 ms) in comparison with the case (avg. 2164 ms) of CLG BN 3, because of the large number (21) of cycles in CLG BN 4. In the case of divergence, the algorithm for CLG BN 2 halted with numeric underflow in the Lambda or Pi value computation. In this case, the algorithm performed with very poor accuracy (avg. 106.35) and had a long execution time (avg. 9678 ms). Figure 6: Percentages for each model case when the algorithm converged, diverged and ran out of time. Some cases for CLG BNs 1, 2 and 4 stopped because the maximum execution time was exceeded. This never happened in CLG BN 3, which contained no cycles. In addition, accuracies for CLG BNs 1 and 2 were better than those for CLG BN 4.