Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Variational Approach in Filtering and Correlated Noise

Sharan Srinivasan, Vijay Gupta, Harsha Honnappa

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.

The Variational Approach in Filtering and Correlated Noise

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.

Paper Structure

This paper contains 13 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The Variational Formulation
  3. Path Measures of Diffusion Processes
  4. A Discrete Time Example
  5. Failure under Correlation
  6. Diffusions with Correlated Noise
  7. Gaussian Case
  8. The General Case
  9. A Variational Formulation for Correlated Noise
  10. Conditional Variational Formula
  11. A Variational Formulation for Linear Diffusions with Correlated Noise
  12. Absolute Continuity and the Free Energy
  13. Conclusion

Key Result

Proposition 2.4

Suppose Assumption as: H1 is satisfied, then for any $y\in\mathbf{Y}$ such that $P_{X|Y}(\cdot,y)$ is the unique minimizer of where $h(P\vert Q)$ is the relative entropy of the measures $P,Q\in\mathcal{P}(\mathcal{X})$, and $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (29)

  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 4.1
  • proof
  • Corollary 4.1.1
  • proof
  • Remark 1
  • Definition 5.1
  • Definition 5.2
  • ...and 19 more