# Optimal Bayesian Affine Estimator and Active Learning for the Wiener Model

arXiv: https://arxiv.org/abs/2504.05490 · HTML: https://mlsanity.com/paper/2504.05490/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 41 references that are in the map.
- First posted on arXiv 7 Apr 2025.

## Builds on (most cited first, 8 of 8)

- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Adaptive Gradient Descent without Descent](https://mlsanity.com/paper/1910.09529/) (2019)
- [Nonlinear System Identification: A User-Oriented Roadmap](https://mlsanity.com/paper/1902.00683/) (2019)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [Identification of block-oriented nonlinear systems starting from linear approximations: A survey](https://mlsanity.com/paper/1607.01217/) (2016)
- [Optimal input design for non-linear dynamic systems: a graph theory approach](https://mlsanity.com/paper/1310.4706/) (2013)

## Built on by (most cited first, 1 of 1)

- [Nonlinear Bayesian Estimator for Parameter Learning: A Fixed-Point Characterization](https://mlsanity.com/paper/2606.10111/) (2026)

## Abstract (arXiv)

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.