Optimal Bayesian Affine Estimator and Active Learning for the Wiener Model
This work addresses parameter estimation in Wiener models, which is important for control and system identification, but it is incremental as it builds on existing Bayesian frameworks with specific enhancements.
The paper tackles the problem of estimating unknown parameters in Wiener models with known linear state dynamics by deriving a closed-form optimal affine estimator using dynamic basis statistics, and it develops an active learning algorithm to minimize estimation error, showing significant improvements over traditional methods in numerical experiments.
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 expressions. Furthermore, we identify an inherent inconsistency in the Fourier bases for single-trajectory measurements, regardless of the 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.