Sequential Monte Carlo algorithms, or particle filters, are widely used for approximating intractable integrals, particularly those arising in Bayesian inference and state-space models. We introduce a new variance reduction technique, the knot operator, which improves the efficiency of particle filters by incorporating potential function information into part, or all, of a transition kernel. The knot operator induces a partial ordering of Feynman-Kac models that implies an order on the asymptotic variance of particle filters, offering a new approach to algorithm design. We discuss connections to existing strategies for designing efficient particle filters, including model marginalisation. Our theory generalises such techniques and provides quantitative asymptotic variance ordering results. We revisit the fully-adapted (auxiliary) particle filter using our theory of knots to show how a small modification guarantees an asymptotic variance ordering for all relevant test functions.

翻译：序贯蒙特卡洛算法，即粒子滤波器，被广泛用于逼近难解积分，特别是在贝叶斯推理和状态空间模型中出现的积分。我们引入了一种新的方差缩减技术——节点算子，通过将势函数信息融入转移核的部分或全部，提升了粒子滤波器的效率。节点算子诱导了费曼-卡克模型的偏序关系，这隐含了粒子滤波器渐近方差的排序，为算法设计提供了新途径。我们讨论了与现有高效粒子滤波器设计策略（包括模型边缘化）的联系。我们的理论推广了这些技术，并提供了定量的渐近方差排序结果。利用节点理论，我们重新审视了完全适应（辅助）粒子滤波器，展示了如何通过微小修改保证所有相关测试函数的渐近方差排序。