A probabilistic Taylor expansion with Gaussian processes

TL;DR

This paper introduces a Gaussian process framework that replicates truncated Taylor expansions through derivative evaluations and Taylor kernels, enabling a probabilistic interpretation of Taylor series with potential for parameter estimation.

Contribution

It presents a novel Gaussian process model that reproduces Taylor expansions using derivative data and analyzes parameter estimation within this framework.

Findings

Posterior mean matches truncated Taylor expansion.

Framework allows maximum likelihood estimation of kernel parameters.

Connects Gaussian process regression with Taylor series approximation.

Abstract

We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.