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arxiv: 2605.20411 · v1 · pith:FYOA6GP6new · submitted 2026-05-19 · 📡 eess.SY · cs.SY

Max-Entropy Moment Filtering for Stochastic Hybrid Systems

Kaito Iwasaki , Tejaswi K. C. , Anthony Bloch , Maani Ghaffari , Taeyoung Lee This is my paper

Pith reviewed 2026-05-21 07:14 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords stochastic hybrid systemsmaximum entropy filteringmoment propagationDynkin's formulaboundary flux correctionreset eventsKalman filternon-Gaussian uncertainty
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The pith

A maximum-entropy moment filter tracks non-Gaussian uncertainty in stochastic hybrid systems by correcting moments for reset events.

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

The paper develops an extension of the Max-Entropy Moment Kalman Filter for stochastic hybrid systems that combine continuous stochastic dynamics with discrete resets. It propagates a finite collection of moments using a rule based on Dynkin's formula that includes a boundary-flux correction for probability jumps across guard sets. This approach allows reconstruction of the belief using maximum-entropy distributions constrained by those moments. A sympathetic reader would care because direct propagation of the full density via hybrid Fokker-Planck equations is computationally expensive, while this method yields tractable dynamics that still capture the non-Gaussian effects from resets.

Core claim

The hybrid MEM-KF performs filtering from partial statistical information by propagating moments through stochastic hybrid dynamics and reconstructing beliefs with moment-constrained maximum-entropy distributions. The key step is a moment propagation rule derived from Dynkin's formula with a jump-sum boundary-flux correction over the guard set. This yields tractable moment dynamics without solving the underlying hybrid PDE. In a stochastic bouncing-ball example, the method captures reset-induced non-Gaussianity through the corrected moment equations.

What carries the argument

The moment propagation rule with jump-sum boundary-flux correction over the guard set, which accounts for reset effects in the moment dynamics derived from Dynkin's formula.

Load-bearing premise

The moment propagation rule derived from Dynkin's formula with a jump-sum boundary-flux correction over the guard set provides a sufficient approximation to the true hybrid Fokker-Planck dynamics for consistent filtering.

What would settle it

Running Monte Carlo simulations of the stochastic bouncing ball system and comparing the moments and reconstructed distributions from the filter against the empirical moments from many trajectories would falsify the claim if significant discrepancies appear in the non-Gaussian features after resets.

Figures

Figures reproduced from arXiv: 2605.20411 by Anthony Bloch, Kaito Iwasaki, Maani Ghaffari, Taeyoung Lee, Tejaswi K. C..

Figure 1
Figure 1. Figure 1: From density evolution to moment-based uncertainty [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of moment dynamics for the stochastic [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heat map of normalized rollout errors for propagated [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Posterior density snapshots for the stochastic bounc [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: True and estimated trajectories for position and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Stochastic hybrid systems combine continuous-time stochastic dynamics with discrete reset events, producing intrinsically non-Gaussian and often multimodal uncertainty. A consistent propagation law must also account for boundary-induced probability flux across guard sets, making direct density propagation through hybrid Fokker-Planck equations expensive. We develop a hybrid extension of the Max-Entropy Moment Kalman Filter (MEM-KF) that performs filtering from partial statistical information by propagating a finite collection of moments through stochastic hybrid dynamics and reconstructing beliefs using moment-constrained maximum-entropy distributions. The key step is a moment propagation rule derived from Dynkin's formula with a jump-sum, in which reset effects appear as a boundary-flux correction over the guard set. This yields tractable moment dynamics without solving the underlying hybrid PDE. In a stochastic bouncing-ball example, the proposed method captures reset-induced non-Gaussianity through corrected moment equations while retaining the MEM-KF's optimization-based maximum-entropy representation.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a hybrid extension of the Max-Entropy Moment Kalman Filter (MEM-KF) for stochastic hybrid systems. It derives a moment propagation rule from Dynkin's formula that incorporates a jump-sum boundary-flux correction over guard sets to account for reset events, yielding tractable moment dynamics without solving the underlying hybrid Fokker-Planck PDE. The approach is illustrated on a stochastic bouncing-ball example, where the corrected moments are claimed to capture reset-induced non-Gaussianity while retaining the optimization-based maximum-entropy belief representation.

Significance. If the central derivation is shown to be consistent, the work would provide a practical moment-based filtering method for hybrid stochastic systems that naturally produces non-Gaussian beliefs, which is relevant for applications such as impact dynamics or switched systems. The use of Dynkin's formula as the starting point is a standard and appropriate tool, and the retention of the max-entropy reconstruction step is a clear strength that avoids ad-hoc density assumptions.

major comments (2)
  1. [Section 3] The moment propagation rule (Section 3, derivation following Dynkin's formula): the jump-sum boundary-flux correction is asserted to close the dynamics using only the tracked moments, but for state-dependent or nonlinear reset maps the flux integral over the guard set generally involves expectations that couple to higher-order moments. No explicit closure assumption or proof that the correction remains within the chosen moment set is provided, which is load-bearing for the claim of consistent filtering.
  2. [Section 4] Stochastic bouncing-ball example (Section 4): the abstract states that the method captures reset-induced non-Gaussianity through the corrected moment equations, yet no quantitative results, error metrics (e.g., moment errors, Wasserstein distance, or comparison to Monte-Carlo ground truth), or validation details are supplied. This absence prevents assessment of whether the boundary-flux term produces accurate propagated moments.
minor comments (2)
  1. Notation for the guard set, reset map, and the precise form of the jump-sum term should be introduced with a dedicated preliminary subsection to improve readability for readers unfamiliar with hybrid stochastic processes.
  2. The abstract would benefit from a brief statement of the number of moments tracked and the optimization problem solved at each filter step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive evaluation of the potential impact of our work on moment-based filtering for stochastic hybrid systems. We provide point-by-point responses to the major comments below and outline the revisions we will make to address them.

read point-by-point responses

  1. Referee: [Section 3] The moment propagation rule (Section 3, derivation following Dynkin's formula): the jump-sum boundary-flux correction is asserted to close the dynamics using only the tracked moments, but for state-dependent or nonlinear reset maps the flux integral over the guard set generally involves expectations that couple to higher-order moments. No explicit closure assumption or proof that the correction remains within the chosen moment set is provided, which is load-bearing for the claim of consistent filtering.

    Authors: We thank the referee for highlighting this important aspect of the derivation. The current manuscript presents the moment propagation rule derived from Dynkin's formula with the boundary-flux correction term expressed as an integral over the guard set. For the class of systems considered, including the linear reset maps in the bouncing-ball example, this term can be evaluated using the available moments when combined with the maximum-entropy reconstruction. However, we agree that a more explicit statement of the closure properties is warranted. In the revised version, we will add a paragraph in Section 3 discussing the conditions under which the correction closes within the moment set (e.g., affine resets and polynomial guard functions) and note that for fully nonlinear cases, moment closure approximations may be employed as in standard moment methods. revision: yes

  2. Referee: [Section 4] Stochastic bouncing-ball example (Section 4): the abstract states that the method captures reset-induced non-Gaussianity through the corrected moment equations, yet no quantitative results, error metrics (e.g., moment errors, Wasserstein distance, or comparison to Monte-Carlo ground truth), or validation details are supplied. This absence prevents assessment of whether the boundary-flux term produces accurate propagated moments.

    Authors: We acknowledge that the example section would benefit from quantitative validation to better demonstrate the effectiveness of the proposed correction. In the revised manuscript, we will augment Section 4 with numerical comparisons against Monte Carlo simulations. Specifically, we will report the errors in the propagated moments (e.g., mean and variance errors) and include a metric such as the Kullback-Leibler divergence or Wasserstein distance between the maximum-entropy reconstructed density and the empirical distribution obtained from a large number of sample paths. This will provide concrete evidence that the boundary-flux term improves the accuracy of the moment propagation. revision: yes

Circularity Check

0 steps flagged

Derivation from standard Dynkin's formula is self-contained

full rationale

The paper's central derivation applies Dynkin's formula (a standard result from stochastic processes) to obtain moment dynamics for hybrid systems, then augments it with an explicit boundary-flux correction term over the guard set to account for resets. This step is presented as a direct consequence of the formula rather than a fit to data or a self-referential definition. No equations reduce to their own inputs by construction, no parameters are fitted and then relabeled as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The resulting moment propagation rule is therefore independent of the target filtering application and remains falsifiable against the underlying hybrid Fokker-Planck equation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard stochastic calculus and the maximum-entropy principle; no free parameters, new entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • standard math Dynkin's formula with jump-sum applies to the stochastic hybrid dynamics and yields the boundary-flux correction
    Invoked to derive the moment propagation rule without solving the hybrid PDE.

pith-pipeline@v0.9.0 · 5701 in / 1130 out tokens · 38403 ms · 2026-05-21T07:14:26.345338+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Spectral bayesian estimation for general Stochastic Hybrid Systems,

    W. Wang and T. Lee, “Spectral bayesian estimation for general Stochastic Hybrid Systems,”Automatica, vol. 117, p. 108989, 2020

    work page 2020

  2. [2]

    A study of the long-term behavior of hybrid systems with symmetries via reduction and the Frobenius–Perron operator,

    M. Oprea, A. Shaw, R. Huq, K. Iwasaki, D. Kassabova, and W. Clark, “A study of the long-term behavior of hybrid systems with symmetries via reduction and the Frobenius–Perron operator,”SIAM Journal on Applied Dynamical Systems, vol. 23, no. 4, pp. 2899–2938, 2024

    work page 2024

  3. [3]

    Uncertainty propagation of stochastic hybrid systems: a case study for types of jump,

    T. K.C., W. Clark, and T. Lee, “Uncertainty propagation of stochastic hybrid systems: a case study for types of jump,”IFAC-PapersOnLine, vol. 59, no. 19, pp. 280–285, 2025, 13th IFAC Symposium on Nonlinear Control Systems NOLCOS 2025

    work page 2025

  4. [4]

    Applebaum,L ´evy Processes and Stochastic Calculus, 2nd ed., ser

    D. Applebaum,L ´evy Processes and Stochastic Calculus, 2nd ed., ser. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2009

    work page 2009

  5. [5]

    A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems,

    J. Bect, “A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems,”Nonlinear Analysis: Hybrid Systems, vol. 4, no. 2, pp. 357–370, 2010, IFAC World Congress 2008

    work page 2010

  6. [6]

    Maximum entropy in the problem of moments,

    L. R. Mead and N. Papanicolaou, “Maximum entropy in the problem of moments,”Journal of Mathematical Physics, vol. 25, no. 8, pp. 2404–2417, 1984

    work page 1984

  7. [7]

    Max Entropy Moment Kalman Filter for polynomial systems with arbitrary noise,

    S. Teng, H. Zhang, D. Jin, A. Jasour, R. Vasudevan, M. Ghaffari, and L. Carlone, “Max Entropy Moment Kalman Filter for polynomial systems with arbitrary noise,” inThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025

    work page 2025

  8. [8]

    Global optimization with polynomials and the problem of moments,

    J. B. Lasserre, “Global optimization with polynomials and the problem of moments,”SIAM Journal on Optimization, vol. 11, no. 3, pp. 796– 817, 2001

    work page 2001

  9. [9]

    Cambridge Texts in Applied Mathematics

    ——,An Introduction to Polynomial and Semi-Algebraic Optimization, ser. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2015

    work page 2015

  10. [10]

    S. M. Kay,Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Prentice Hall, 1993

    work page 1993

  11. [11]

    A maximum entropy approach to the moment closure problem for Stochastic Hybrid Systems at equilibrium,

    J. Zhang, L. DeVille, S. Dhople, and A. D. Dom ´ınguez-Garc´ıa, “A maximum entropy approach to the moment closure problem for Stochastic Hybrid Systems at equilibrium,” in53rd IEEE Conference on Decision and Control, 2014, pp. 747–752

    work page 2014

  12. [12]

    Estimation of non-normalized statistical models by score matching,

    A. Hyv ¨arinen, “Estimation of non-normalized statistical models by score matching,”J. Mach. Learn. Res., vol. 6, pp. 695–709, 2005

    work page 2005