Variational Gaussian approximation of the Kushner optimal filter

TL;DR

This paper introduces a variational Gaussian approximation method for the Kushner optimal filter, providing a unified framework that generalizes Kalman-Bucy and Riccati flows to nonlinear systems using Wasserstein and Fisher metrics.

Contribution

It proposes a novel variational approach to approximate the Kushner equation with Gaussian flows, extending linear filtering techniques to nonlinear cases.

Findings

The method recovers Kalman-Bucy and Riccati flows in linear scenarios.

It provides a tractable approximation for nonlinear filtering problems.

The approach unifies Wasserstein and Fisher metric-based updates.

Abstract

In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one.