# Bee Swarm Metropolis-Hastings Sampling for Bayesian Inference in the Ginzburg-Landau Equation
## 5 Conclusions and Future Work  
In this study, we proposed the BeeSwarm-MH algorithm, which integrates BSO with the MH acceptance criterion, effectively overcoming the limitations of traditional Bayesian inference sampling methods. In parameter estimation experiments for the GL equation, the proposed algorithm reduced runtime by 65.3% compared to the classical MH method, achieving convergence with only 10% of the iterations required by MH. Key parameter estimation biases were below 0.001, and all R-hat convergence diagnostics were below 1.0014, satisfying Bayesian inference convergence criteria. The core innovation of the algorithm lies in the novel integration of BSO-inspired scout-worker bee division of labor with the MH acceptance mechanism, which is supplemented by a two-level adaptive step size adjustment strategy. This design achieves a dynamic \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Metric** & **Classical MH** & **BeeSwarm-MH** & **Classical MH 95\% CI** & **BeeSwarm-MH 95\% CI** \\ \hline Acceptance Rate & 0.3731 & 0.3771 & - & - \\ Effective Sample Size (ESS) & [2447, 3154, 15, 830] & [1451, 1237, 2208] & - & - \\ \hline \(\beta_{1}\) Mean & 0.9601 & 0.9597 & [0.9244, 0.9969] & [0.9244, 0.9968] \\ \(\gamma_{0}\) Mean & 0.9865 & 0.9869 & [0.9595, 1.0133] & [0.9608, 1.0134] \\ \(\beta_{0}\) Mean & \(-\)3.0286 & \(-\)3.0293 & [\(-\)3.0782, \(-\)2.9796] & [\(-\)3.0751, -2.9807] \\ \hline R-hat Statistic & [1.0000, 1.0000, 1.0002] & [1.0001, 1.0014, 1.0003] & All \(<1.1\) (Convergence Criterion) \\ Model \(R^{2}\) & 0.9424 & Comparable Fit Quality \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of sampling quality and parameter estimates. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Feature** & **Classical MH** & **BeeSwarm-MH** \\ \hline Exploration Mechanism & Random walk, local search & Swarm cooperation, global search \\ Adaptivity & Fixed step size, manual tuning & Dynamic step size and swarm structure \\ Convergence Speed & Full iteration required & Early convergence (1/10 iterations) \\ Parameter Tuning Difficulty & High (manual proposal tuning) & Low (automatic optimization) \\ \hline \hline \multirow{2}{*}{Suitable Scenarios} & Simple models, high precision samples, & Complex models, limited computation \\  & well-known parameter spaces & resources, unknown multimodal \\ \multicolumn{3}{c}{parameter spaces} \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of algorithmic characteristics and applicability. balance between global exploration and local exploitation, addressing the sensitivity of classical MH algorithms to step size selection. Future work will focus on three main directions: extending the algorithm's applicability to high-dimensional parameter spaces by enhancing parallel computing efficiency; conducting an in-depth theoretical analysis of convergence rates and ergodicity under multimodal target distributions; and applying the algorithm to complex physical systems, biomedical modeling, and real-time inference scenarios. Specifically, we will pursue GPU-accelerated implementations, derive spectral-gap bounds via weak-Poincare techniques, and release an open-source Python package for real-world deployment. Additionally, challenges related to extremely multimodal distributions and strong parameter dependencies warrant further investigation to improve robustness and scalability. Conceptualization, S.X. and L.Z. ; methodology, L.Z. ; software, L.Z. ; validation, S.X. ; formal analysis, L.Z. ; investigation, S.X. ; resources, S.X. and L.Z. ; writing--original draft preparation, S.X. ; writing--review and editing, L.Z. ; visualization, S.X. ; supervision, L.Z. ; project administration, L.Z. ; funding acquisition, S.X. and L.Z. All authors have read and agreed to the published version of the manuscript. Supported by the Open Fund of Zhejiang Key Laboratory of Film and TV Media Technology, No. 2024E10023.