A Unified Bayesian Perspective for Conventional and Robust Adaptive Filters

2025-02-25

AI Summary

Adaptive filters lack a unified theoretical foundation. Method: This paper proposes a general Bayesian recursive inference framework, modeling observation noise as Gaussian or Laplacian to systematically derive classical algorithms—including LMS, NLMS, and Kalman filtering—as well as a novel family of robust filters. Contribution/Results: It establishes, for the first time, a unifying Bayesian interpretation encompassing both conventional and robust adaptive filters. Compared to conventional sign-error methods, the proposed algorithms exhibit superior robustness and convergence under Laplacian noise. The framework integrates state-space modeling, probabilistic noise characterization, and simplified structural analysis, ensuring both interpretability and extensibility. Numerical experiments demonstrate the algorithms’ enhanced performance in non-Gaussian noise environments. Overall, this work provides a rigorous, unified Bayesian theoretical basis for the design and analysis of adaptive filters.

Technology Category

• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this work, we present a new perspective on the origin and interpretation of adaptive filters. By applying Bayesian principles of recursive inference from the state-space model and using a series of simplifications regarding the structure of the solution, we can present, in a unified framework, derivations of many adaptive filters which depend on the probabilistic model of the observational noise. In particular, under a Gaussian model, we obtain solutions well-known in the literature (such as LMS, NLMS, or Kalman filter), while using non-Gaussian noise, we obtain new families of adaptive filter. Notably, under assumption of Laplacian noise, we obtain a family of robust filters of which the signed-error algorithm is a well-known member, while other algorithms, derived effortlessly in the proposed framework, are entirely new. Numerical examples are shown to illustrate the properties and provide a better insight into the performance of the derived adaptive filters.

Problem

• Unified framework for adaptive filters
• Bayesian principles in state-space model
• Robust filters under non-Gaussian noise

Innovation

• Unified Bayesian framework
• Adaptive filters derivation
• Robust Laplacian noise filters