# Wiener Chaos in Kernel Regression: Towards Untangling Aleatoric and Epistemic Uncertainty

arXiv: https://arxiv.org/abs/2312.07387 · HTML: https://mlsanity.com/paper/2312.07387/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 11 of 11)

- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Data-driven polynomial chaos expansion for machine learning regression](https://mlsanity.com/paper/1808.03216/) (2018)
- [Posterior and Computational Uncertainty in Gaussian Processes](https://mlsanity.com/paper/2205.15449/) (2022)
- [Posterior Variance Analysis of Gaussian Processes with Application to Average Learning Curves](https://mlsanity.com/paper/1906.01404/) (2019)
- [Sharp Calibrated Gaussian Processes](https://mlsanity.com/paper/2302.11961/) (2023)
- [Equilibrium Distributions and Stability Analysis of Gaussian Process State Space Models](https://mlsanity.com/paper/1811.06653/) (2018)
- [Behavioral Theory for Stochastic Systems? A Data-driven Journey from Willems to Wiener and Back Again](https://mlsanity.com/paper/2209.06414/) (2022)
- [On a Stochastic Fundamental Lemma and Its Use for Data-Driven Optimal Control](https://mlsanity.com/paper/2111.13636/) (2021)

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

- [Towards safe Bayesian optimization with Wiener kernel regression](https://mlsanity.com/paper/2411.02253/) (2024)

## Abstract (arXiv)

Gaussian Processes (GPs) are a versatile method that enables different approaches towards learning for dynamics and control. Gaussianity assumptions appear in two dimensions in GPs: The positive semi-definite kernel of the underlying reproducing kernel Hilbert space is used to construct the co-variance of a Gaussian distribution over functions, while measurement noise (i.e. data corruption) is usually modeled as i.i.d. additive Gaussians. In this note, we generalize the setting and consider kernel ridge regression with additive i.i.d. non-Gaussian measurement noise. To apply the usual kernel trick, we rely on the representation of the uncertainty via polynomial chaos expansions, which are series expansions for random variables of finite variance introduced by Norbert Wiener. We derive and discuss the analytic L2 solution to the arising Wiener kernel regression. Considering a polynomial dynamic system as a numerical example, we show that our approach allows us to distinguish the uncertainty that stems from the noise in the data samples from the total uncertainty encoded in the GP posterior distribution.