Revisiting the Two-Filter Formula for Smoothing for State-Space Models

TL;DR

This paper revisits the two-filter smoothing formula for state-space models, demonstrating its effectiveness and extensions for linear, Gaussian, and non-Gaussian models, including particle filters and Gaussian-sum smoothing.

Contribution

It provides a comprehensive analysis of the two-filter smoothing formula, including its application to non-Gaussian models and particle filters, with practical insights on optimal smoothing methods.

Findings

Similar posterior distributions can be obtained with proper inverse filter design.

Gaussian-sum smoothing works for high-dimensional non-Gaussian models with Gaussian-mixture noise.

Fixed lag smoothing often yields better results than the two-filter formula in particle filtering.

Abstract

Smoothing algorithms for state-space models, i.e., fixed-interval smoothing, fixed-lag smoothing, and two-filter formula for smoothing, are examined using real examples. For linear and Gaussian state-space models, it is observed that similar posterior distributions can be obtained by properly defining the inverse filter. In the case of linear non-Gaussian state-space models, it is shown that Gaussian-sum smoothing is possible even for relatively high dimensional state-space model with Gaussian-mixture noise inputs by properly setting the inverse filter. The two-filter formula is also applicable for particle filter, but better results are obtained with fixed lag smoothing or with the average of forward and backward fixed lag smoothers.