The Bootstrap Particle Filter (BPF) and the Ensemble Kalman Filter (EnKF) are two widely used methods for sequential Bayesian filtering: the BPF is asymptotically exact but can suffer from weight degeneracy, while the EnKF scales well in high dimension yet is exact only in the linear-Gaussian case. We combine these approaches by retaining the EnKF transport step and adding a principled importance-sampling correction. Our first contribution is a general importance-sampling theory for mixture targets and proposals, including variance comparisons between individual- and mixture-based estimators. We then interpret the stochastic EnKF analysis as sampling from explicit Gaussian-mixture proposals obtained by conditioning on the current or previous ensemble, which leads to six self-normalized IS-EnKF schemes. We embed these updates into a broader class of ensemble-based filters and prove consistency and error bounds, including weight-variance comparisons and sufficient conditions ensuring finite-variance importance weights. As a second contribution, we construct transported quasi-Monte Carlo (TQMC) point sets for the Gaussian-mixture laws arising in prediction and analysis, yielding TQMC-enhanced variants that can substantially reduce sampling error without changing the filtering pipeline. Numerical experiments on benchmark models compare the proposed mixture-weighted and TQMC-enhanced filters, showing improved filtering accuracy relative to BPF, EnKF, and the standard weighted EnKF, and that the weighted schemes eliminate the EnKF error plateau often caused by analysis-target mismatch.

翻译：Bootstrap粒子滤波（BPF）与集成卡尔曼滤波（EnKF）是两种广泛使用的序贯贝叶斯滤波方法：BPF具有渐近精确性但可能面临权重退化问题，而EnKF在高维情况下扩展性良好，但仅在线性高斯情形下精确。我们通过保留EnKF的传输步骤并添加一种基于原理的重要性采样校正，将这两种方法相结合。我们的第一个贡献是针对混合目标分布与建议分布的一般重要性采样理论，包括基于个体与基于混合的估计量之间的方差比较。随后，我们将随机EnKF分析过程解释为从通过当前或先前集成条件化获得的显式高斯混合建议分布中采样，从而推导出六种自归一化IS-EnKF方案。我们将这些更新嵌入更广泛的基于集成的滤波器类别中，并证明其一致性与误差界，包括权重方差比较以及确保重要性权重具有有限方差的充分条件。作为第二个贡献，我们为预测与分析中出现的高斯混合分布构造了传输拟蒙特卡洛（TQMC）点集，从而得到TQMC增强变体，这些变体能在不改变滤波流程的前提下显著降低采样误差。在基准模型上的数值实验比较了所提出的混合加权与TQMC增强滤波器，结果显示相对于BPF、EnKF及标准加权EnKF，其滤波精度得到提升，且加权方案消除了常由分析目标失配引起的EnKF误差平台。