Particle smoothing enables state estimation in nonlinear and non-Gaussian state-space models, but its practical use is often limited by high computational cost. Backward smoothing methods such as the Forward Filter Backward Smoother (FFBS) and its marginal form (FFBSm) can achieve high accuracy, yet typically require quadratic computational complexity in the number of particles. This paper examines the accuracy--computational cost trade-offs of particle smoothing methods through a trend-estimation example. Fixed-lag smoothing, FFBS, and FFBSm are compared under Gaussian and heavy-tailed (Cauchy-type) system noise, with particular attention to O(m) approximations of FFBSm based on subsampling and local neighborhood restrictions. The results show that FFBS and FFBSm outperform fixed-lag smoothing at a fixed particle number, while fixed-lag smoothing often achieves higher accuracy under equal computational time. Moreover, efficient FFBSm approximations are effective for Gaussian transitions but become less advantageous for heavy-tailed dynamics.

翻译：粒子平滑方法能够实现非线性非高斯状态空间模型中的状态估计，但其实际应用常受限于高昂的计算成本。后向平滑方法，如前向滤波后向平滑器（FFBS）及其边际形式（FFBSm），虽能达到较高精度，但通常需要粒子数量的二次计算复杂度。本文通过一个趋势估计示例，深入探讨了粒子平滑方法在精度与计算成本之间的权衡。在系统噪声分别为高斯分布和重尾（柯西型）分布的条件下，比较了固定滞后平滑、FFBS 和 FFBSm 的性能，并特别关注了基于子采样和局部邻域限制的 FFBSm 的 O(m) 近似方法。结果表明，在固定粒子数条件下，FFBS 和 FFBSm 优于固定滞后平滑；而在相等计算时间条件下，固定滞后平滑通常能达到更高的精度。此外，高效的 FFBSm 近似方法对于高斯状态转移是有效的，但对于重尾动态系统则优势减弱。