The Variational Approach in Filtering and Correlated Noise

Harsha Honnappa; Sharan Srinivasan; Vijay Gupta

arxiv: 2604.03001 · v1 · submitted 2026-04-03 · 🧮 math.PR · cs.SY· eess.SY· math.OC

The Variational Approach in Filtering and Correlated Noise

Sharan Srinivasan , Vijay Gupta , Harsha Honnappa This is my paper

Pith reviewed 2026-05-13 18:10 UTC · model grok-4.3

classification 🧮 math.PR cs.SYeess.SYmath.OC

keywords variational filteringnonlinear filteringcorrelated noisediffusion processesrelative entropyfree energyMitter-Newton formulationsingular measures

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The pith

The Mitter-Newton variational filter fails for correlated signal-observation noises because the joint and product measures are mutually singular.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The variational formulation of nonlinear filtering minimizes a free energy that includes relative entropy with respect to a product reference measure. This formulation requires the joint path measure of signal and observation to be absolutely continuous with respect to that product measure. When the two diffusions share a common noise source the measures become mutually singular, so no reference measure works. The paper replaces the product reference with a conditional one that keeps the correlation structure intact. The new principle recovers the original result when noises are independent and supplies an explicit free-energy description of the filter for linear correlated-noise systems.

Core claim

The absolute continuity condition between the joint path measure and any product reference measure necessarily fails whenever signal and observation diffusions share a common noise source, rendering the measures mutually singular. The authors therefore introduce a conditional variational principle whose reference measure preserves the noise correlation. This yields a free-energy characterization of the filter that reduces to the Mitter-Newton formulation precisely when the noises are independent and produces an explicit expression in the linear correlated-noise case.

What carries the argument

A conditional variational principle that substitutes a correlation-preserving reference measure for the usual product reference measure.

If this is right

When the noises are independent the conditional principle reduces exactly to the original Mitter-Newton free-energy minimization.
In the linear case with correlated noise the filter is characterized explicitly as the minimizer of the new free-energy functional.
The filtering distribution remains the unique minimizer of the free-energy functional under the generalized reference measure.
No choice of product reference measure can restore the original formulation once correlation is present.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach opens the door to variational filtering in sensor networks where environmental disturbances induce correlation between signal and measurement noise.
It suggests that similar conditional reference measures could be constructed for other classes of processes whose increments are dependent.
Numerical implementations would replace the singular product reference with a joint simulation that respects the shared noise.

Load-bearing premise

The joint path measure of the signal and observation must be absolutely continuous with respect to some product reference measure.

What would settle it

An explicit pair of linear diffusions driven by a shared Brownian motion for which the Radon-Nikodym derivative between the joint law and any product measure is shown to be undefined or infinite almost surely.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows the Mitter-Newton variational filter breaks for shared noise sources because the joint and product measures are mutually singular, then fixes it with a conditional variational principle that keeps the correlation and gives an explicit linear case.

read the letter

The central point is that the usual variational characterization of the filter, which minimizes a free-energy functional against a product reference measure, cannot work when signal and observation noises are correlated. The paper proves the joint path measure and any product reference are mutually singular because quadratic covariation is a path functional that is nonzero under the joint law but zero under the product law. They replace the prior with a reference that retains the correlation structure, recover the independent-noise case as a special instance, and derive an explicit free-energy form for the linear correlated setting. This is a clean, targeted adjustment rather than a routine extension. The argument uses standard stochastic calculus and measure theory with no free parameters or self-referential steps. The only real limitation right now is that we have only the abstract, so the full details of the singularity proof and the new principle cannot be checked for any hidden gaps in the technical steps. Still, the logic as stated shows no internal contradiction. This is for people working on stochastic filtering and control who run into correlated noise in applications. A reader already familiar with the Mitter-Newton setup will see the value in the linear characterization and the generalization. It is worth sending to a serious referee because the problem is concrete, the fix is motivated by the failure of the old approach, and the claims are stated in a way that can be verified formally.

Referee Report

2 major / 2 minor

Summary. The paper shows that the Mitter-Newton variational formulation of nonlinear filtering, which minimizes a free-energy functional involving relative entropy to a product reference measure, fails when the signal and observation diffusions share a driving noise. The authors prove mutual singularity of the joint path measure and any product reference measure by observing that quadratic covariation is a measurable functional of paths, taking a nonzero value under the joint law but zero under the product law. They introduce a conditional variational principle that replaces the prior with a reference measure preserving the noise correlation structure. This recovers the original formulation for independent noises and supplies an explicit free-energy characterization of the filter in the linear correlated-noise case.

Significance. If the central claims hold, the work is significant because it removes a structural obstruction to applying variational methods in filtering to the correlated-noise regime that arises in most applications. The singularity proof is a direct, parameter-free application of stochastic calculus, and the conditional principle yields a concrete, usable characterization in the linear setting without introducing new free parameters. These features strengthen the variational approach as a practical and theoretically sound tool.

major comments (2)

[Main singularity theorem] The proof that quadratic covariation distinguishes the joint and product measures (central to the singularity claim) must explicitly verify that the functional remains measurable and separates the measures under the precise regularity conditions imposed on the diffusion coefficients; this step is load-bearing for the motivation of the new principle.
[Linear-case section] In the linear correlated-noise characterization, the construction of the correlation-preserving reference measure must be shown to yield a well-defined minimization problem whose unique minimizer is the filter; any implicit dependence on the unknown filter would undermine the variational principle.

minor comments (2)

[Introduction and notation] The notation distinguishing the original product reference measure from the new correlation-preserving reference measure should be introduced earlier and used consistently throughout the statements of the theorems.
[Recovery of independent case] A brief remark on how the conditional variational principle reduces exactly to the Mitter-Newton functional when the cross-variation term vanishes would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments on our manuscript. We address each major comment below and will incorporate the requested clarifications and expansions into the revised version.

read point-by-point responses

Referee: [Main singularity theorem] The proof that quadratic covariation distinguishes the joint and product measures (central to the singularity claim) must explicitly verify that the functional remains measurable and separates the measures under the precise regularity conditions imposed on the diffusion coefficients; this step is load-bearing for the motivation of the new principle.

Authors: We agree that the measurability of the quadratic covariation functional and its separating property must be verified explicitly under the precise regularity conditions (Lipschitz continuity and linear growth) imposed on the diffusion coefficients. In the revised manuscript we will add a dedicated lemma in the proof of the main singularity theorem. The lemma will establish that the map sending a continuous path to its quadratic covariation process is measurable with respect to the Borel sigma-algebra generated by the uniform topology, and that this functional takes a strictly positive value almost surely under the joint law while remaining zero almost surely under any product reference measure. The argument relies only on the given coefficient assumptions and standard properties of stochastic integrals, thereby making the singularity claim fully rigorous and strengthening the motivation for the conditional variational principle. revision: yes
Referee: [Linear-case section] In the linear correlated-noise characterization, the construction of the correlation-preserving reference measure must be shown to yield a well-defined minimization problem whose unique minimizer is the filter; any implicit dependence on the unknown filter would undermine the variational principle.

Authors: We appreciate this observation. The correlation-preserving reference measure is constructed explicitly from the known diffusion coefficients and the fixed correlation structure of the driving Brownian motions; its definition does not involve the filtering distribution. In the revised manuscript we will expand the linear-case section to include: (i) an explicit construction of the reference measure via Girsanov transformation that preserves the quadratic covariation, (ii) a proof that the associated free-energy functional is strictly convex and lower semicontinuous on the space of probability measures absolutely continuous with respect to the reference, and (iii) a verification that its unique minimizer coincides with the Kalman filter (which is known to be the unique conditional distribution). This establishes that the variational problem is well-posed without any implicit dependence on the unknown filter. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins with a standard stochastic calculus fact: quadratic covariation is a measurable path functional that is nonzero under the joint law but zero under any product reference measure, implying mutual singularity. This is an external mathematical result, not a self-definition or fitted input. The conditional variational principle is then introduced by direct replacement of the reference measure to retain the observed correlation structure, recovering the Mitter-Newton case when noises are independent. No equation or claim reduces to the paper's own inputs by construction, and no self-citation is load-bearing.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard assumptions from stochastic calculus for diffusion processes and the definition of path measures; no free parameters or invented entities are introduced.

axioms (1)

domain assumption Signal and observation processes are Itô diffusions possibly driven by correlated Brownian motions.
Standard modeling choice in nonlinear filtering theory invoked to set up the joint path measure.

pith-pipeline@v0.9.0 · 5442 in / 1220 out tokens · 43706 ms · 2026-05-13T18:10:02.925917+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

[1]

Andrew L Allan, Jost Pieper, and Josef Teichmann,Rough SDEs and robust filtering for jump-diffusions, arXiv preprint arXiv:2507.05930 (2025)

work page internal anchor Pith review arXiv 2025
[2]

62, American Mathematical Soc., 1998

Vladimir Igorevich Bogachev,Gaussian measures, no. 62, American Mathematical Soc., 1998

work page 1998
[3]

J. M. C. Clark,The design of robust approximations to the stochastic differential equations of nonlinear filtering, Communication Systems and Random Process Theory25(1978), 721–734

work page 1978
[4]

1, 43–56

John MC Clark and Dan Crisan,On a robust version of the integral representation formula of nonlinear filtering, Probability theory and related fields133(2005), no. 1, 43–56

work page 2005
[5]

Crisan, J

D. Crisan, J. Diehl, P. K. Friz, and H. Oberhauser,Robust filtering: Correlated noise and multidimen- sional observation, The Annals of Applied Probability23(2013), no. 5, 2139–2160

work page 2013
[6]

Davis and Michael P

Mark H.A. Davis and Michael P. Spathopoulos,Pathwise nonlinear filtering for nondegenerate diffusions with noise correlation, SIAM Journal on Control and Optimization25(1987), no. 2, 260–278

work page 1987
[7]

4, 559–578

Robert J Elliott and Michael Kohlmann,Robust filtering for correlated multidimensional observations, Mathematische Zeitschrift178(1981), no. 4, 559–578

work page 1981
[8]

Xi Geng,An introduction to the theory of rough paths, Lecture Notes (2021), 9

work page 2021
[9]

5, 1813–1833

Sanjoy K Mitter and Nigel J Newton,A variational approach to nonlinear estimation, SIAM journal on control and optimization42(2003), no. 5, 1813–1833

work page 2003
[10]

02, 2350016

Huijie Qiao,Convergence of nonlinear filtering for multiscale systems with correlated L´ evy noises, Stochastics and Dynamics23(2023), no. 02, 2350016

work page 2023
[11]

3, 902–918

Huijie Qiao and Jinqiao Duan,Nonlinear filtering of stochastic dynamical systems with L´ evy noises, Advances in Applied Probability47(2015), no. 3, 902–918. THE V ARIATIONAL APPROACH IN FILTERING AND CORRELATED NOISE 13

work page 2015
[12]

Sharan Srinivasan, Vijay Gupta, and Harsha Honnappa,Robust filtering of l´ evy-driven stochastic models, arXiv preprint arXiv:2602.14310 (2026). (S. Srinivasan)Elmore F amily School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA Email address:srini256@purdue.edu (V. Gupta)Elmore F amily School of Electrical and Computer...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

Andrew L Allan, Jost Pieper, and Josef Teichmann,Rough SDEs and robust filtering for jump-diffusions, arXiv preprint arXiv:2507.05930 (2025)

work page internal anchor Pith review arXiv 2025

[2] [2]

62, American Mathematical Soc., 1998

Vladimir Igorevich Bogachev,Gaussian measures, no. 62, American Mathematical Soc., 1998

work page 1998

[3] [3]

J. M. C. Clark,The design of robust approximations to the stochastic differential equations of nonlinear filtering, Communication Systems and Random Process Theory25(1978), 721–734

work page 1978

[4] [4]

1, 43–56

John MC Clark and Dan Crisan,On a robust version of the integral representation formula of nonlinear filtering, Probability theory and related fields133(2005), no. 1, 43–56

work page 2005

[5] [5]

Crisan, J

D. Crisan, J. Diehl, P. K. Friz, and H. Oberhauser,Robust filtering: Correlated noise and multidimen- sional observation, The Annals of Applied Probability23(2013), no. 5, 2139–2160

work page 2013

[6] [6]

Davis and Michael P

Mark H.A. Davis and Michael P. Spathopoulos,Pathwise nonlinear filtering for nondegenerate diffusions with noise correlation, SIAM Journal on Control and Optimization25(1987), no. 2, 260–278

work page 1987

[7] [7]

4, 559–578

Robert J Elliott and Michael Kohlmann,Robust filtering for correlated multidimensional observations, Mathematische Zeitschrift178(1981), no. 4, 559–578

work page 1981

[8] [8]

Xi Geng,An introduction to the theory of rough paths, Lecture Notes (2021), 9

work page 2021

[9] [9]

5, 1813–1833

Sanjoy K Mitter and Nigel J Newton,A variational approach to nonlinear estimation, SIAM journal on control and optimization42(2003), no. 5, 1813–1833

work page 2003

[10] [10]

02, 2350016

Huijie Qiao,Convergence of nonlinear filtering for multiscale systems with correlated L´ evy noises, Stochastics and Dynamics23(2023), no. 02, 2350016

work page 2023

[11] [11]

3, 902–918

Huijie Qiao and Jinqiao Duan,Nonlinear filtering of stochastic dynamical systems with L´ evy noises, Advances in Applied Probability47(2015), no. 3, 902–918. THE V ARIATIONAL APPROACH IN FILTERING AND CORRELATED NOISE 13

work page 2015

[12] [12]

Sharan Srinivasan, Vijay Gupta, and Harsha Honnappa,Robust filtering of l´ evy-driven stochastic models, arXiv preprint arXiv:2602.14310 (2026). (S. Srinivasan)Elmore F amily School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA Email address:srini256@purdue.edu (V. Gupta)Elmore F amily School of Electrical and Computer...

work page internal anchor Pith review Pith/arXiv arXiv 2026