The two filter formula reconsidered: Smoothing in partially observed Gauss--Markov models without information parametrization
This work addresses a technical bottleneck in statistical inference for researchers in signal processing and state estimation, offering incremental improvements to existing methods.
The paper tackles the problem of smoothing in partially observed Gauss-Markov models by re-examining the two filter formula, replacing the traditional information parametrization with a recursion over log-quadratic likelihoods, which simplifies the square-root formulation and provides formulae for forward Markov representation.
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.