⚡ electrical engineering

Max-Entropy Moment Filtering for Stochastic Hybrid Systems

This paper proposes a hybrid extension of the Max-Entropy Moment Kalman Filter that enables efficient state estimation for stochastic hybrid systems by deriving a moment propagation rule with boundary-flux corrections via Dynkin's formula, thereby capturing non-Gaussian reset effects without solving expensive hybrid partial differential equations.

Original authors: Kaito Iwasaki, Tejaswi K. C., Anthony Bloch, Maani Ghaffari, Taeyoung Lee

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kaito Iwasaki, Tejaswi K. C., Anthony Bloch, Maani Ghaffari, Taeyoung Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict where a bouncy ball will be in the next few seconds. But this isn't a normal ball; it's a "chaotic" ball. It moves smoothly through the air, but it's also being pushed around by invisible, random gusts of wind (stochastic noise). Worse yet, every time it hits the floor, it doesn't just bounce back normally; it gets a sudden, jarring "reset" that changes its speed instantly, often in a way that makes its future path very hard to guess.

This is the problem the authors tackle: How do you track the location of something that moves smoothly, gets hit by random noise, and suddenly jumps to a new state when it hits a wall?

Here is the breakdown of their solution, using simple analogies.

The Problem: The "Too Complicated" Map

Usually, when scientists try to track moving objects, they assume the object follows a nice, predictable bell curve (like a Gaussian distribution). Think of this as a smooth, symmetrical hill where the peak is the most likely spot.

But in a "Stochastic Hybrid System" (like our bouncing ball), the rules break that smooth hill.

The Noise: Random wind makes the ball drift.
The Reset: When the ball hits the floor (the "guard"), it instantly changes velocity. This creates a weird, lopsided, or even multi-peaked shape in the probability map. It's no longer a smooth hill; it's a jagged, crumpled piece of paper.

To track this perfectly, you would normally need to solve a massive, complex equation (the Fokker-Planck equation) that maps out every single possible shape of that crumpled paper. The authors say this is like trying to draw a perfect, high-definition map of a storm cloud in real-time. It's too expensive and slow to do on a computer.

The Solution: The "Snapshot" Strategy

Instead of trying to map the entire crumpled paper (the full probability density), the authors decided to track only a few key statistical snapshots (called "moments").

Think of these moments like describing a cloud by its:

Center: Where is the middle of the cloud?
Width: How spread out is it?
Skew: Is it leaning to the left or right?
Peaks: Does it have a second bump?

By tracking just these numbers (the "moments"), they avoid the heavy math of mapping the whole cloud.

The Secret Sauce: The "Max-Entropy" Guess

Here is the tricky part: If you only know the center and width of a cloud, there are infinite ways the cloud could actually look. It could be a smooth hill, or it could be two separate hills. The math doesn't tell you which one it is. This is called the "closure problem."

To solve this, the authors use a principle called Maximum Entropy.

The Analogy: Imagine you are a detective who only knows the suspect's height and weight. You don't know their face. To make the fairest guess possible without inventing fake details, you assume the suspect looks like the "most average" person who fits those stats. You don't guess they have a mustache or a hat unless the data forces you to.
In the paper: They take the few moments they calculated and ask, "What is the most 'unbiased' or 'random' shape that fits these numbers?" This gives them a best-guess probability distribution that is mathematically sound but doesn't invent fake patterns.

The "Jump" Correction

The hardest part of the bouncing ball is the moment it hits the floor. The math for smooth movement is well-known, but the "jump" breaks the rules.

The authors developed a special rule (based on something called Dynkin's Formula) that acts like a traffic cop at the boundary.

When the ball approaches the floor, the math calculates how much "probability traffic" is flowing toward the wall.
When it hits, the rule instantly redirects that traffic. It takes the probability of the ball hitting the floor and "teleports" it to the new post-bounce speed.
This allows them to update their "moment snapshots" correctly without having to simulate the actual crash.

The Result: A Smarter Filter

They tested this on a bouncing ball with random wind.

Low-Order Guess (Standard Method): If you only track the center and width (like a standard GPS), the filter thinks the ball is always in a nice, smooth bell curve. When the ball hits the floor and creates a weird, lopsided shape, this method fails to see it.
Their Method (High-Order): By tracking more complex "snapshots" (skewness, peaks, etc.) and using the Max-Entropy rule, their filter could see the ball developing a "lopsided" shape after hitting the floor. It matched the reality of the chaotic ball much better than standard methods.

Summary

The paper presents a new way to track chaotic, bouncing objects. Instead of trying to draw the entire, messy picture of where the object might be, they track a few key numbers (moments). When the object hits a wall and jumps, they use a special "traffic cop" math rule to update those numbers. Finally, they use a "fairness principle" (Max-Entropy) to fill in the gaps and guess the most likely shape of the object's location. This is much faster and more accurate for bouncy, chaotic systems than trying to map every single possibility.

Technical Summary: Max-Entropy Moment Filtering for Stochastic Hybrid Systems

Problem Statement
Stochastic hybrid systems (SHSs) combine continuous-time stochastic dynamics with discrete reset events. These systems, common in mechanical impacts, robotic contact, and biological switching, inherently generate non-Gaussian and often multimodal uncertainty. Traditional filtering approaches face two primary challenges:

Non-Gaussianity: Reset events induce complex probability distributions that standard Gaussian filters (like the Kalman Filter) cannot represent.
Computational Intractability: Direct propagation of probability densities via hybrid Fokker-Planck equations requires solving partial differential equations (PDEs) with complex boundary conditions (guard sets), which is computationally expensive and difficult to apply to filtering from partial statistical information.

The paper addresses the problem of estimating the state of an SHS from noisy measurements by propagating uncertainty using only partial statistical information (moments) rather than full density functions.

Methodology
The authors propose a Max-Entropy Moment Kalman Filter (MEM-KF) adapted for stochastic hybrid systems. The methodology consists of three core components:

Moment Propagation via Dynkin's Formula:
Instead of evolving probability densities, the method evolves a finite collection of moments (expected values of observables). The authors derive a hybrid extension of Dynkin's formula that accounts for both continuous diffusion and discrete jumps.
- The continuous evolution is governed by the infinitesimal generator of the stochastic Koopman operator.
- The discrete reset effects are incorporated as a boundary-flux correction over the guard set $G$ .
- The resulting evolution equation for a moment $m_\alpha(t)$ is:
  $\frac{d}{dt}m_\alpha(t) = \mathbb{E}[(\mathcal{A}\phi_\alpha)(X_t)] + \int_G (\phi_\alpha(\Delta(x)) - \phi_\alpha(x))(J(t, x) \cdot n(x))^+ dS(x)$
  where $\mathcal{A}$ is the generator, $\Delta$ is the reset map, and the integral term represents the probability flux leaving the domain and re-entering via the reset. This allows for tractable moment dynamics without solving the underlying hybrid PDE.
Belief Reconstruction via Maximum Entropy:
Since a finite set of moments does not uniquely determine a probability distribution, the authors employ the Moment-Constrained Maximum Entropy (MED) principle. At each time step, the filter reconstructs the probability density $p^*(x)$ by maximizing entropy subject to the currently propagated moment constraints. This yields a density in the exponential family:
$p^*(x; \lambda) = \frac{1}{Z(\lambda)} \exp\left(-\sum_\alpha \lambda_\alpha x^\alpha\right)$
The parameters $\lambda$ are determined by solving a convex optimization problem (minimizing the dual thermal dynamics potential).
Filtering Algorithm:
The proposed filter operates in a predict-update cycle:
- Prediction: Moments are propagated forward in time using the derived moment ODEs. The boundary flux terms are approximated using the MED reconstructed from the previous time step.
- Update: Upon receiving a measurement, the filter updates the belief. Assuming a polynomial measurement model, the measurement residual is modeled as a moment-constrained MED. Bayes' rule is applied at the moment level, resulting in a new posterior MED.
- State Estimation: A point estimate (MAP) is obtained by solving a polynomial optimization problem over the feasible domain.

Key Contributions

Hybrid Moment Dynamics: Derivation of a consistent moment propagation rule for SHSs using Dynkin's formula with a jump-sum, where reset effects appear explicitly as boundary flux corrections.
Tractable Filtering: A framework that avoids solving hybrid Fokker-Planck equations by working exclusively with finite-dimensional moment vectors and convex reconstruction.
Non-Gaussian Representation: The ability to capture reset-induced non-Gaussianity and multimodality through higher-order moments and MED reconstruction, overcoming the limitations of linear-Gaussian estimators.

Results
The method was validated on a stochastic bouncing-ball example (a single-mode SHS with gravity, drag, noise, and velocity resets upon impact).

Moment Accuracy: The propagated moments (up to order $r=4$ ) showed close agreement with Monte Carlo reference simulations.
Density Reconstruction: MED reconstructions with $r=4$ successfully captured the skewness and multimodal structure of the velocity distribution caused by impacts. In contrast, a second-order truncation ( $r=2$ ) forced a Gaussian (unimodal) reconstruction, failing to represent the impact-induced deformation.
State Estimation: In a state estimation task with intermittent, non-Gaussian noisy position measurements, the filter accurately tracked the true state trajectory, achieving a root-mean-square error (RMSE) of $6.91 \times 10^{-2}$ m for position and $8.62 \times 10^{-1}$ m/s for velocity.

Significance and Claims
The paper claims that this framework provides a principled and tractable approach to uncertainty propagation in SHSs. By combining moment propagation with maximum entropy, the method captures essential hybrid features (non-Gaussianity, multimodality) that are lost in standard linear estimators, while remaining computationally more feasible than direct density propagation.

The authors acknowledge limitations and future directions, noting that:

The method relies on the choice of truncation order; selecting a representation rich enough to retain hybrid features in high dimensions remains an open challenge.
Computational costs increase with dimension and order due to the need to repeatedly solve convex optimization problems and evaluate partition functions.
The approach assumes a specific structure (single mode in the example, though generalizable) and relies on polynomial observables and reset maps.

The work positions itself as a step toward scalable filtering for complex stochastic hybrid systems, bridging the gap between rigorous density-based theory and practical, moment-based estimation.