# Bee Swarm Metropolis-Hastings Sampling for Bayesian Inference in the Ginzburg-Landau Equation
## 4 Bayesian Inference for the Ginzburg-Landau Equation
### Experimental Results and Analysis for BeeSwarm-MH Inference  
Based on the Bayesian inference framework utilizing the BeeSwarm-MH sampling, we conduct an in-depth study of the method's performance and statistical significance in parameter estimation through both graphical visualization and algorithmic analysis. Figure 6 shows the MCMC trace plots for the parameters \(\beta_{1}\), \(\gamma_{0}\), and \(\beta_{0}\), where the red dashed lines indicate the burn-in cutoff. It can be observed that all three chains rapidly enter a stable fluctuation region after burn-in, exhibiting no apparent systematic drift or significant trending behavior. This indicates that the Markov chains have successfully converged to the target posterior distributions and demonstrate good mixing properties. The sampling traces maintain an approximately stable mean level amidst high-frequency oscillations, further confirming adequate exploration of the parameter space and providing a reliable foundation for posterior statistical inference. Figure 7 presents the marginal posterior distributions of the corresponding parameters. All parameters exhibit unimodal characteristics: \(\gamma_{0}\) appears approximately symmetric, \(\beta_{1}\) shows slight right skewness, and \(\beta_{0}\) displays slight left skewness. The black dashed lines represent the posterior means, while the gray dashed lines denote the 95% posterior credible intervals, illustrating the concentration of estimates and uncertainty bounds. Overall, the posterior distributions are smooth with well-defined mean estimates, reflecting strong parameter identifiability given the observed data and demonstrating the robustness and statistical reliability of the posterior inference. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Number of Samples (\(n_{\text{samples}}\)) & 100,000 \\ Swarm Size (\(n_{\text{bees}}\)) & 12 \\ Scout Bee Ratio (\(\rho\)) & 0.5 \\ Base Proposal Standard Deviation (\(\sigma_{0}\)) & \([0.01,0.01,0.1]\) \\ Global Scaling Factor (\(c\)) & 1.0 (initial value) \\ Burn-In Period & 30\% of total iterations \\ Convergence Check Interval & Every 5000 iterations \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Hyperparameter values for BeeSwarm-MH algorithm. Figure 8 depicts the joint posterior relationship matrix for the parameters \(\beta_{1}\), \(\gamma_{0}\), and \(\beta_{0}\). This figure provides a multidimensional characterization of parameter dependencies via univariate marginal distributions on the diagonal, scatter plots above the diagonal, and kernel density contour plots below the diagonal. On the diagonal, both \(\beta_{1}\) and \(\gamma_{0}\) display unimodal, concentrated distributions, while \(\beta_{0}\) exhibits a clear peak with well-defined intervals, indicating robust univariate posterior estimates. The upper-triangular scatter plots reveal a weak positive correlation clustering trend between \(\beta_{1}\) and \(\gamma_{0}\), whereas other variable pairs show more dispersed relationships, exposing complex dependence structures. The lower-triangular contour plots further confirm that the joint distribution of \(\beta_{1}\) and \(\gamma_{0}\) is concentrated along a specific direction, while \(\beta_{0}\) shows multimodal local clustering with the other two parameters, highlighting nontrivial nonlinear associations. Table 5 reports the parameter-specific posterior means and the overall Gelman-Rubin \(\hat{R}\) statistic computed from four independent MCMC chains after 10,000 iterations (burn-in 3000). All three parameters achieve \(\hat{R}<1.1\), confirming adequate convergence. Figure 6: MCMC trace plots for BeeSwarm-MH sampling. Figure 7: Posterior distributions of GL-equation parameters obtained via BeeSwarm-MH. Figure 9 displays the post-burn-in trace plots (left) and the corresponding chain-wise Gelman-Rubin \(\hat{R}\) statistics (right) for the three model parameters \(\beta_{1}\), \(\gamma_{0}\), and \(\beta_{0}\).