Nonlinear Estimator: Dual Bayesian Affine Estimators for Parameter Learning

2026-06-08

AI Summary

This work addresses the challenge of nonlinear parameter estimation in Wiener-type state-space models by proposing a fixed-point iteration-based dual estimator framework. The approach couples two affine minimum mean square error (MMSE) estimators to separately handle unknown parameters and latent states, while introducing dynamic basis statistics (DBS) to efficiently summarize information from nonlinear basis functions. The resulting dual state-parameter and dual basis-parameter estimators alternately update their prior information, enabling stable and efficient nonlinear learning. Extensive Monte Carlo experiments demonstrate that the dual state-parameter estimator significantly outperforms existing methods—including purely affine estimators, particle Gibbs, and expectation-maximization (EM) variants of sequential Monte Carlo algorithms—in terms of parameter mean square error.

Abstract

This paper presents a nonlinear parameter estimator for Wiener-type state-space models obtained as a fixed-point architecture that couples two affine minimum mean-squared error (MMSE) estimators: one for the unknown parameters and one for latent variables. The architecture retains the functional structure of the optimal affine MMSE parameter estimator while incorporating Dynamic Basis Statistics (DBS) estimates that summarize nonlinear basis-function evaluations. Two DBS construction strategies are developed, leading to two nonlinear estimator frameworks. The dual basis-parameter estimator combines an affine basis estimator with the affine parameter estimator, whereas the dual state-parameter estimator first computes affine state estimates and their covariances, then maps these state-estimate statistics through a Gaussian DBS operator to obtain DBS estimates. Both dual estimators admit fixed-point characterizations that alternate between estimating each component using the updated prior of the other, obtained from that component's plug-in estimate statistics from the previous iteration. The efficacy of the proposed methods is examined via extensive Monte Carlo experiments, showing that the dual basis-parameter estimator attains parameter mean-squared errors comparable to those of the purely affine parameter estimator, while the dual state-parameter estimator achieves the lowest parameter mean-squared error, outperforming both the dual basis-parameter and purely affine parameter estimators, as well as sequential Monte Carlo variants of classical Particle Gibbs and Expectation-Maximization schemes.

Problem

• nonlinear estimation
• parameter learning
• state-space models
• Wiener systems
• MMSE estimation

Innovation

• Nonlinear Estimation
• Dual Bayesian Affine Estimators
• Dynamic Basis Statistics
• Fixed-Point Architecture
• Wiener-type State-Space Models