When performing Bayesian inference using Sequential Monte Carlo (SMC) methods, two considerations arise: the accuracy of the posterior approximation and computational efficiency. To address computational demands, Sequential Monte Carlo Squared (SMC$^2$) is well-suited for high-performance computing (HPC) environments. The design of the proposal distribution within SMC$^2$ can improve accuracy and exploration of the posterior as poor proposals may lead to high variance in importance weights and particle degeneracy. The Metropolis-Adjusted Langevin Algorithm (MALA) uses gradient information so that particles preferentially explore regions of higher probability. In this paper, we extend this idea by incorporating second-order information, specifically the Hessian of the log-target. While second-order proposals have been explored previously in particle Markov Chain Monte Carlo (p-MCMC) methods, we are the first to introduce them within the SMC$^2$ framework. Second-order proposals not only use the gradient (first-order derivative), but also the curvature (second-order derivative) of the target distribution. Experimental results on synthetic models highlight the benefits of our approach in terms of step-size selection and posterior approximation accuracy when compared to other proposals.

翻译：在使用序列蒙特卡洛（SMC）方法进行贝叶斯推断时，需兼顾两个考量：后验近似的准确性与计算效率。为应对计算需求，平方序列蒙特卡洛（SMC$^2$）方法非常适合高性能计算（HPC）环境。SMC$^2$中提议分布的设计可提升后验探索的准确性与效率，因为低质量的提议可能导致重要性权重的高方差与粒子退化问题。Metropolis-Adjusted Langevin Algorithm（MALA）利用梯度信息使粒子优先探索高概率区域。本文通过引入二阶信息——具体为对数目标函数的Hessian矩阵——对此思路进行拓展。虽然二阶提议在粒子马尔可夫链蒙特卡洛（p-MCMC）方法中已有研究，但我们是首次将其引入SMC$^2$框架。二阶提议不仅利用目标分布的梯度（一阶导数），还利用其曲率（二阶导数）。在合成模型上的实验结果表明，相较于其他提议方法，我们的方法在步长选择与后验近似精度方面具有显著优势。