The particle filter (PF), also known as the sequential Monte Carlo (SMC), is designed to approximate high-dimensional probability distributions and their normalizing constants in the discrete-time setting. To reduce the variance of the Monte Carlo approximation, several twisted particle filters (TPF) have been proposed by researchers, where one chooses or learns a twisting function that modifies the Markov transition kernel. In this paper, we study the TPF from a continuous-time perspective. Under suitable settings, we show that the discrete-time model converges to a continuous-time limit, which can be solved through a series of well-studied control-based importance sampling algorithms. This discrete-continuous connection allows the design of new TPF algorithms inspired by established continuous-time algorithms. As a concrete example, guided by existing importance sampling algorithms in the continuous-time setting, we propose a novel algorithm called ``Twisted-Path Particle Filter" (TPPF), where the twist function, parameterized by neural networks, minimizes specific KL-divergence between path measures. Some numerical experiments are given to illustrate the capability of the proposed algorithm.

翻译：粒子滤波器（PF），亦称为序列蒙特卡罗（SMC），旨在离散时间设定下近似高维概率分布及其归一化常数。为降低蒙特卡罗近似的方差，研究者们提出了多种扭曲粒子滤波器（TPF），其通过选择或学习一个扭曲函数来修改马尔可夫转移核。本文从连续时间的视角研究TPF。在适当的设定下，我们证明离散时间模型收敛于一个连续时间极限，该极限可通过一系列经过充分研究的基于控制的重要性采样算法求解。这种离散-连续联系使得能够借鉴成熟的连续时间算法来设计新的TPF算法。作为一个具体示例，在现有连续时间设定下重要性采样算法的指导下，我们提出了一种称为“扭曲路径粒子滤波器”（TPPF）的新算法，其中由神经网络参数化的扭曲函数最小化路径测度间的特定KL散度。文中给出了一些数值实验以说明所提算法的性能。