Gaussian filtering and variational approximations for Bayesian smoothing in continuous-discrete stochastic dynamic systems

TL;DR

This paper compares Gaussian filtering-based smoothing and variational Gaussian smoothing for nonlinear stochastic systems, showing the variational approach can iteratively improve results, especially in highly nonlinear cases.

Contribution

It introduces a variational Gaussian smoother, compares it with traditional filtering-based methods, and extends the approach to systems with singular diffusion matrices.

Findings

Variational Gaussian smoother outperforms filtering-based methods in nonlinear systems.

Linearization and sigma-point methods effectively approximate intractable expectations.

The approach extends to systems with singular diffusion matrices.

Abstract

The Bayesian smoothing equations are generally intractable for systems described by nonlinear stochastic differential equations and discrete-time measurements. Gaussian approximations are a computationally efficient way to approximate the true smoothing distribution. In this work, we present a comparison between two Gaussian approximation methods. The Gaussian filtering based Gaussian smoother uses a Gaussian approximation for the filtering distribution to form an approximation for the smoothing distribution. The variational Gaussian smoother is based on minimizing the Kullback-Leibler divergence of the approximate smoothing distribution with respect to the true distribution. The results suggest that for highly nonlinear systems, the variational Gaussian smoother can be used to iteratively improve the Gaussian filtering based smoothing solution. We also present linearization and sigma-point methods to approximate the intractable Gaussian expectations in the Variational Gaussian smoothing equations. In addition, we extend the variational Gaussian smoother for certain class of systems with singular diffusion matrix.