Log-Normalization Constant Estimation using the Ensemble Kalman-Bucy Filter with Application to High-Dimensional Models

TL;DR

This paper develops ensemble Kalman-Bucy filter-based methods for estimating the log-normalization constant in continuous-time filtering models, providing theoretical error bounds and applying the approach to high-dimensional linear and nonlinear models.

Contribution

It introduces new bias analysis and error bounds for ensemble Kalman-Bucy filter estimates of normalization constants, extending their application to high-dimensional filtering and parameter estimation.

Findings

Error bounds depend on time horizon and ensemble size

Method performs well for both linear and nonlinear models

Provides theoretical guarantees for online parameter estimation

Abstract

In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman-Bucy filter based estimates based upon several nonlinear Kalman-Bucy diffusions. Based upon new conditional bias results for the mean of the afore-mentioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_n-$errors and conditional bias. Depending on the type of nonlinear Kalman-Bucy diffusion, we show that these are of order $(\sqrt{t/N}) + t/N$ or $1/\sqrt{N}$ ($\mathbb{L}_n-$errors) and of order $[t+\sqrt{t}]/N$ or $1/N$ (conditional bias), where $t$ is the time horizon and $N$ is the ensemble size. Finally, we use these results for online static parameter estimation for above filtering models and implement the methodology for both linear and nonlinear models.