# Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks +
## 5 Experiment  
This section presents experiments to evaluate the performance of the HMP-GMR algorithm and the Optimal GMR algorithm. For evaluation of the HMP-GMR algorithm, [PERSON] et al. [28] presented simple experiments to demonstrate scalability and efficiency using two hybrid BNs. Here, more extensive experiments are performed on four BNs. These four BNs would be representative BNs in terms of various numbers of discrete parent nodes and various numbers of loopy structures in a given network. Figure 2 shows two illustrative Conditional Gaussian (CG) BNs (i.e., 1 and 2) containing a discrete node \(A\) with 4 states, a continuous node \(X_{j}\) and another continuous node \(Y_{j}\). These two BNs differ in the links between \(Y_{j}\) and \(Y_{j+1}\). The first, shown in Figure 1(a), has no undirected cycles involving only continuous nodes, while the second, shown in Figure 1(b), has undirected cycles among. For example, between \(X_{1}\) and \(Y_{2}\), there are two paths: \(X_{1}\to X_{2}\to Y_{2}\) and \(X_{1}\to Y_{1}\to Y_{2}\). Figure 3 shows two additional cases (Figure 2(a),b) in which the BNs contain a large number of discrete nodes. Each discrete node \(A_{i}\) has four states and the continuous nodes \(X_{i}\) and \(Y_{j}\) are conditional Gaussians. Note that Figure 2(a) has no undirected cycles involving only continuous nodes, while Figure 2(b) has loopy structure for the continuous nodes. The experiments examined both CLG and CNG BNs with these four CG BNs. The size of the four CG BNs was varied by adjusting \(n\). Some of the leaf continuous nodes \(\{Y_{1},\,\ldots,\,Y_{n}\}\) were randomly selected as evidence nodes. All other nodes were unobserved. Table 2 shows characteristics of these four CG BNs. In these BNs, the discrete nodes have four states. Therefore, in CG BNs 1 and 2, there are four discrete states for discrete node \(A\), while in CG BNs 3 and 4, there are \(4^{n}\) total configurations of the discrete states. This is \(4^{7}\) = 16384 configurations when \(n\) = 7. When \(n\) = 7, CG BNs 1 and 4 contain 21 cycles, while CG BN 2 contains 501 cycles (Cycles were derived by a cycle finding algorithm [41]). CG BN 3 contains no cycles. Figure 2: Conditional Gaussian BNs. The following factors were varied in the experiment: (1) Type of hybrid BN (i.e., BN 1, 2, 3 or 4; CLG or CNG BN), (2) type of inference algorithm (i.e., Hybrid Junction Tree (Hybrid-JT) [2,3], original Hybrid MP (HMP) [29], Hybrid MP with Gaussian Mixture Reduction (HMP-GMR) [28] or Likelihood Weighting (LW) sampling [10]), (3) number of repeated structures \(n\), (4) algorithm characteristics (i.e., the number of GMM components allowed, the number of allowable message passing iterations and the maximum precision for the message passing algorithms). The dependent variables were accuracy of result and execution time. For all experiments, the convergence criterion for HMP and HMP-GMR was \(max\_prs=10^{-3}\). Using these settings, we conducted three experiments: (1) A comparison between HMP, HMP-GMR and Hybrid-JT investigated scalability of HMP-GMR to complex networks (i.e., larger values of \(n\)), (2) the HMP-GMR itself was evaluated for posterior distribution accuracy and execution time and (3) optimal settings derived from the HMP-GMR-OS algorithm were evaluated by inference accuracy. The experiments were run on a 3.40 GHz Intel Core i7-3770 processor. The algorithms were implemented in the Java programming language (The source codes are available online at \"[[https://sourceforge.net/p/prognos/code/HEAD/tree/trunk/cps2/](https://sourceforge.net/p/prognos/code/HEAD/tree/trunk/cps2/)]([https://sourceforge.net/p/prognos/code/HEAD/tree/trunk/cps2/](https://sourceforge.net/p/prognos/code/HEAD/tree/trunk/cps2/))\"). In a Java code for HMP-GMR, there was another exit point from the iteration of the HMP-GMR algorithm in Section 3. In some cases, a computation between Lambda values and Pi values in the HMP-GMR algorithm could not be computed due to numeric underflow. This happened when the HMP-GMR algorithm diverged.