The two filter formula reconsidered: Smoothing in partially observed Gauss--Markov models without information parametrization

TL;DR

This paper re-examines the two filter formula in Gauss--Markov models, proposing a likelihood-based recursion that simplifies the algorithm and removes the need for information parametrization.

Contribution

It introduces a likelihood-based recursion for the two filter formula, avoiding information form and simplifying the square-root algorithm.

Findings

Likelihood recursion replaces traditional information form

Simplifies the square-root filtering algorithm

Provides formulas for forward Markov representation

Abstract

In this article, the two filter formula is re-examined in the setting of partially observed Gauss--Markov models. It is traditionally formulated as a filter running backward in time, where the Gaussian density is parametrized in ``information form''. However, the quantity in the backward recursion is strictly speaking not a distribution, but a likelihood. Taking this observation seriously, a recursion over log-quadratic likelihoods is formulated instead, which obviates the need for ``information'' parametrization. In particular, it greatly simplifies the square-root formulation of the algorithm. Furthermore, formulae are given for producing the forward Markov representation of the a posteriori distribution over paths from the proposed likelihood representation.