# Bee Swarm Metropolis-Hastings Sampling for Bayesian Inference in the Ginzburg-Landau Equation
###### Abstract  
To improve the sampling efficiency of Markov Chain Monte Carlo in complex parameter spaces, this paper proposes an adaptive sampling method that integrates a swarm intelligence mechanism called the BeeSwarm-MH algorithm. The method combines global exploration by scout bees with local exploitation by worker bees. It employs multi-stage perturbation intensities and adaptive step-size tuning to enable efficient posterior sampling. Focusing on Bayesian inference for parameter estimation in the soliton solutions of the two-dimensional complex Ginzburg-Landau equation, we design a dedicated inference framework to systematically compare the performance of BeeSwarm-MH with the classical Metropolis-Hastings algorithm. Experimental results demonstrate that BeeSwarm-MH achieves comparable estimation accuracy while significantly reducing the required number of iterations and total computation time for convergence. Moreover, it exhibits superior global search capabilities and adaptive features, offering a practical approach for efficient Bayesian inference in complex physical models. Metropolis-Hastings sampling; bee swarm algorithm; Ginzburg-Landau model; Bayesian parameter inference +
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autocorrelation and reduced sampling efficiency [10]. The need for a \"burn-in\" period and strong Markov dependence further exacerbate estimation bias under limited computational resources [11]. To mitigate these challenges, methods such as Adaptive MH [12], Hamiltonian Monte Carlo [4; 13], and Parallel Tempering [14; 15] have been developed to enhance sampling efficiency and robustness. Contemporary Bayesian practice likewise demands a scalable MCMC capable of navigating high-dimensional, multimodal posteriors. A recent survey [16] underscores the growing call for adaptive, parallel, and globally informed sampling strategies. Bee Swarm Optimization (BSO) is a typical swarm intelligence algorithm that mimics the cooperative foraging behavior of bees, balancing global exploration and local exploitation through the role-based division of labor. Scout bees and worker bees collectively conduct parallel and collaborative searches, effectively escaping local optima while sharing information [17; 18]. Thanks to its simple structure, efficient implementation, and ease of parallelization, BSO has found broad applications in continuous and combinatorial optimization as well as industrial engineering [19; 20; 21; 22]. However, BSO is fundamentally designed for optimization rather than sampling, and it lacks rigorous theoretical guarantees for asymptotic convergence or distributional consistency [23]. Without acceptance-rate control, its update mechanism can produce redundant or biased samples, making direct application to Bayesian inference problematic [24]. Empirical studies show that BSO suffers rapid diversity loss and mode collapse as dimension or problem hardness increases [19; 23], confirming its tendency to bias the sampling distribution. Recent work further explores hybrids that weave swarm-based exploration into MH acceptance schemes; irreversible Langevin samplers [25], for example, have been shown to accelerate convergence in rugged posteriors. Notably, no existing method has embedded bee-swarm principles within the MH framework while preserving exactness--a gap this paper addresses by introducing BeeSwarm-MH, which is a novel algorithm of our own design. Given the complementary strengths of MH and BSO, this paper proposes a hybrid approach--the BeeSwarm-MH algorithm--that integrates these advantages. The method retains MH's acceptance criterion to ensure asymptotic correctness while employing BSO-inspired role mechanisms for multi-scale, parallel search. Scout bees perform global exploration to avoid local traps, while worker bees focus on local refinement.