Convolutional Bayesian Filtering: A Generalized Framework for Handling Model Mismatch in State Estimation

Core Concepts

By conditioning on an additional event that stipulates an inequality constraint between the real and virtual states, the standard conditional probabilities in Bayesian filtering can be transformed into convolutional forms. This generalized framework allows for explicit consideration of model mismatch, leading to more robust state estimation algorithms.

Abstract

The content discusses a new framework called "convolutional Bayesian filtering" that extends the standard Bayesian filtering approach to handle model mismatch. The key insights are:

Convolutional conditional probability: By adding an inequality condition between the real and virtual states/measurements, the standard conditional probabilities can be transformed into a convolutional form. This relaxes the need for complete information about the conditional probabilities.

Uncertain hidden Markov model: The real system is modeled as an "uncertain hidden Markov model" that distinguishes between the real and virtual states/measurements, and bounds their differences using the inequality conditions.

Convolutional Bayesian filtering: By substituting the standard total probability rule and Bayes' law with their convolutional counterparts, a generalized filtering framework is established that can explicitly account for model mismatch.

Analytical solution for Gaussian case: When the distance metrics are quadratic forms and the threshold distributions are exponential, an analytical form of convolutional Bayesian filtering can be derived, leading to a robust Kalman filter.

Exponential density rescaling: For non-Gaussian systems, an approximation technique based on exponential density rescaling is proposed, which relates to the information bottleneck theory.

The new framework encompasses standard Bayesian filtering as a special case, and allows for more nuanced handling of model uncertainties, leading to improved state estimation performance.

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Key Insights Distilled From

Convolutional Bayesian Filtering

by Wenhan Cao,S... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00481.pdf

Deeper Inquiries

How can the convolutional Bayesian filtering framework be extended to handle time-varying model mismatch, where the bounds on the real and virtual states/measurements evolve dynamically

To extend the convolutional Bayesian filtering framework to handle time-varying model mismatch, where the bounds on the real and virtual states/measurements evolve dynamically, we can introduce a dynamic threshold parameter that adapts to the changing model uncertainties. This dynamic threshold can be updated at each time step based on the evolving characteristics of the system and the discrepancies between the nominal models and the actual system behavior. By incorporating time-varying thresholds into the convolutional conditional probabilities, we can ensure that the filtering process remains robust and adaptive to the changing model uncertainties over time.

What are the potential applications of convolutional Bayesian filtering beyond state estimation, such as in decision-making or control problems where model uncertainty is a key concern

Convolutional Bayesian filtering has a wide range of potential applications beyond state estimation. One key application is in decision-making problems where model uncertainty plays a significant role. By incorporating convolutional Bayesian filtering into decision-making algorithms, we can account for the uncertainties in the system models and make more informed and robust decisions. Additionally, in control problems, convolutional Bayesian filtering can be used to design adaptive control strategies that can adjust to varying model uncertainties in real-time. This adaptive control approach can enhance the stability and performance of control systems in the presence of dynamic model mismatch.

The information bottleneck theory is used to provide a theoretical justification for the exponential density rescaling technique. Are there other information-theoretic perspectives that could offer additional insights into the convolutional Bayesian filtering approach

While the information bottleneck theory provides a theoretical justification for the exponential density rescaling technique in convolutional Bayesian filtering, there are other information-theoretic perspectives that could offer additional insights into this approach. One such perspective is the concept of mutual information maximization, where the goal is to maximize the mutual information between the observed data and the latent variables while minimizing the mutual information between the latent variables and the model parameters. By framing convolutional Bayesian filtering as a mutual information maximization problem, we can gain a deeper understanding of how the rescaling technique impacts the information flow in the filtering process and how it helps in handling model uncertainties effectively.