In this article we consider the application of multilevel Monte Carlo, for the estimation of normalizing constants. In particular we will make use of the filtering algorithm, the ensemble Kalman–Bucy filter (EnKBF), which is an N-particle representation of the Kalman–Bucy filter (KBF). The EnKBF is of interest as it coincides with the optimal filter in the continuous-linear setting, i.e. the KBF. This motivates our particular setup in the linear setting. The resulting methodology we will use is the multilevel ensemble Kalman–Bucy filter (MLEnKBF). We provide an analysis based on deriving \({\mathbb {L}}_q\)-bounds for the normalizing constants using both the single-level, and the multilevel algorithms, which is largely based on previous work deriving the MLEnKBF Chada et al. (2022). Our results will be highlighted through numerical results, where we firstly demonstrate the error-to-cost rates of the MLEnKBFs comparing it to the EnKBF on a linear Gaussian model. Our analysis will be specific to one variant of the MLEnKBF, whereas the numerics will be tested on different variants. We also exploit this methodology for parameter estimation, where we test this on the models arising in atmospheric sciences, such as the stochastic Lorenz 63 and 96 model.

在本文中，我们考虑应用多级蒙特卡罗来估计归一化常数。特别是我们将使用滤波算法，集成卡尔曼-布西滤波器 (EnKBF)，它是卡尔曼-布西滤波器 (KBF) 的N粒子表示。EnKBF 很有趣，因为它与连续线性设置中的最佳滤波器（即 KBF）一致。这激发了我们在线性设置中的特定设置。我们将使用的最终方法是多级集成卡尔曼-布西滤波器 (MLEnKBF)。我们提供基于推导\({\mathbb {L}}_q\)的分析- 使用单级和多级算法的归一化常数的界限，这在很大程度上基于以前推导 MLEnKBF Chada 等人的工作。（2022 年）。我们的结果将通过数值结果突出显示，我们首先在线性高斯模型上展示 MLEnKBF 与 EnKBF 的误差成本率。我们的分析将针对 MLEnKBF 的一种变体，而数字将在不同的变体上进行测试。我们还利用这种方法进行参数估计，在大气科学中出现的模型上进行测试，例如随机 Lorenz 63 和 96 模型。