The Variational Approach in Filtering and Correlated Noise

TL;DR

This paper analyzes the variational formulation of nonlinear filtering, showing its limitations with correlated noise and proposing a generalized approach that accounts for noise correlation structures.

Contribution

It proves the failure of the existing variational formulation under correlated noise and introduces a new conditional variational principle that generalizes the original.

Findings

The original formulation fails when signal and observation share noise sources.

The new approach preserves noise correlation structure in the variational formulation.

Explicit free energy characterization of the filter in linear correlated-noise cases.

Abstract

The variational formulation of nonlinear filtering due to Mitter and Newton characterizes the filtering distribution as the unique minimizer of a free energy functional involving the relative entropy with respect to the prior and an expected energy. This formulation rests on an absolute continuity condition between the joint path measure and a product reference measure. We prove that this condition necessarily fails whenever the signal and observation diffusions share a common noise source. Specifically we show that the joint and product measures are mutually singular, so no choice of reference measure can salvage the formulation. We then introduce a conditional variational principle that replaces the prior with a reference measure that preserves the noise correlation structure. This generalization recovers the Mitter--Newton formulation as a special case when the noises are independent, and yields an explicit free energy characterization of the filter in the linear correlated-noise setting.