Optimal Bayesian Affine Estimator and Active Learning for the Wiener Model

2025-04-07

AI Summary

This paper addresses parameter estimation for Wiener systems—comprising a known linear dynamic block followed by an unknown nonlinear output function—under the challenging single-trajectory measurement setting, where inherent inconsistency arises due to insufficient excitation. We propose a Bayesian-inspired closed-form affine estimator, introducing Dynamic Basis Statistics (DBS) as a novel characterization to achieve unbiased, analytically tractable optimal estimation. Theoretically, we characterize the intrinsic estimation bias induced by persistent excitation deficiency in single-trajectory data. Leveraging this insight, we design an active learning input synthesis algorithm based on Fourier bases that explicitly minimizes the posterior estimation error. The method enables efficient closed-form computation without iterative optimization. Numerical experiments demonstrate that our approach achieves significantly higher estimation accuracy compared to regularized least squares, particularly under limited-data regimes.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This paper presents a Bayesian estimation framework for Wiener models, focusing on learning nonlinear output functions under known linear state dynamics. We derive a closed-form optimal affine estimator for the unknown parameters, characterized by the so-called"dynamic basis statistics (DBS)."Several features of the proposed estimator are studied, including Bayesian unbiasedness, closed-form posterior statistics, error monotonicity in trajectory length, and consistency condition (also known as persistent excitation). In the special case of Fourier basis functions, we demonstrate that the closed-form description is computationally available, as the Fourier DBS enjoys explicit expression. Furthermore, we identify an inherent inconsistency in single-trajectory measurements, regardless of input excitation. Leveraging the closed-form estimation error, we develop an active learning algorithm synthesizing input signals to minimize estimation error. Numerical experiments validate the efficacy of our approach, showing significant improvements over traditional regularized least-squares methods.

Problem

• Bayesian estimation of Wiener model nonlinear functions
• Closed-form affine estimator with dynamic basis statistics
• Active learning algorithm to minimize estimation error

Innovation

• Bayesian affine estimator with dynamic basis statistics
• Closed-form posterior statistics for Fourier basis
• Active learning algorithm minimizing estimation error