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

Filip Tronarp

TL;DR

The two filter formula is re-examined in the setting of partially observed Gauss--Markov models, and formulae are given for producing the forward Markov representation of the a posteriori distribution over paths from the proposed likelihood 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.

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

TL;DR

The two filter formula is re-examined in the setting of partially observed Gauss--Markov models, and formulae are given for producing the forward Markov representation of the a posteriori distribution over paths from the proposed likelihood 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.

Paper Structure

This paper contains 10 sections, 7 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. The new backward-forward recursion
  4. The backward recursion
  5. The forward recursion
  6. The backward-forward recursion in Cholesky form
  7. The two-filter formula
  8. Relationship to information parametrization, maximum likelihood estimation, and the likelihood as a density
  9. An example of unknown origin
  10. Discussion

Key Result

Theorem 1

Consider the partially observed Markov process in eq:pomp, then $h_{t:T \mid t}$ satisfy the following recursion: and the marginal likelihood, $L_{1:T}$ is given by Additionally, the posterior over paths is given by

Figures (2)

  • Figure 1: The position of the object (black), the position observations (red), and a $\pm 2\sigma$ credible interval of the position (blue). The estimate was obtained by the forward-backward algorithm.
  • Figure 2: The position of the object (black), the position observations (red), and a $\pm 2\sigma$ confidence interval of the position (blue). The estimate was obtained by maximum likelihood.

Theorems & Definitions (11)

  • Theorem 1: The backward-forward method
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Remark 1
  • proof
  • ...and 1 more