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

arXiv: https://arxiv.org/abs/2502.21116 · HTML: https://mlsanity.com/paper/2502.21116/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 24 references that are in the map.
- First posted on arXiv 28 Feb 2025.

## Builds on (most cited first, 3 of 3)

- [Temporal Parallelization of Bayesian Smoothers](https://mlsanity.com/paper/1905.13002/) (2019)
- [A New Smoothing Algorithm for Jump Markov Linear Systems](https://mlsanity.com/paper/2004.08561/) (2020)
- [Computing the matrix exponential and the Cholesky factor of a related finite horizon Gramian](https://mlsanity.com/paper/2310.13462/) (2023)

## Abstract (arXiv)

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.