# Nonlinear Bayesian Estimator for Parameter Learning: A Fixed-Point Characterization

arXiv: https://arxiv.org/abs/2606.10111 · HTML: https://mlsanity.com/paper/2606.10111/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 53 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 12 of 12)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Elements of Sequential Monte Carlo](https://mlsanity.com/paper/1903.04797/) (2019)
- [On Markov chain Monte Carlo methods for tall data](https://mlsanity.com/paper/1505.02827/) (2015)
- [Particle Gibbs with Ancestor Sampling](https://mlsanity.com/paper/1401.0604/) (2014)
- [Nested Sequential Monte Carlo Methods](https://mlsanity.com/paper/1502.02536/) (2015)
- [Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty](https://mlsanity.com/paper/2003.02359/) (2020)
- [Identification of block-oriented nonlinear systems starting from linear approximations: A survey](https://mlsanity.com/paper/1607.01217/) (2016)
- [MCMC-Based Inference in the Era of Big Data: A Fundamental Analysis of the Convergence Complexity of High-Dimensional Chains](https://mlsanity.com/paper/1508.00947/) (2015)
- [On backward smoothing algorithms](https://mlsanity.com/paper/2207.00976/) (2022)
- [Variational System Identification for Nonlinear State-Space Models](https://mlsanity.com/paper/2012.05072/) (2020)
- [Optimal Bayesian Affine Estimator and Active Learning for the Wiener Model](https://mlsanity.com/paper/2504.05490/) (2025)

## Abstract (arXiv)

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.