# A probabilistic Taylor expansion with Gaussian processes

arXiv: https://arxiv.org/abs/2102.00877 · HTML: https://mlsanity.com/paper/2102.00877/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 51 references that are in the map.
- First posted on arXiv 1 Feb 2021.

## Builds on (most cited first, 10 of 10)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [Bayesian Optimization with Gradients](https://mlsanity.com/paper/1703.04389/) (2017)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Scaling Gaussian Process Regression with Derivatives](https://mlsanity.com/paper/1810.12283/) (2018)
- [Probabilistic ODE Solvers with Runge-Kutta Means](https://mlsanity.com/paper/1406.2582/) (2014)
- [Probabilistic Interpretation of Linear Solvers](https://mlsanity.com/paper/1402.2058/) (2014)
- [A Bayes-Sard Cubature Method](https://mlsanity.com/paper/1804.03016/) (2018)
- [Probabilistic Linear Multistep Methods](https://mlsanity.com/paper/1610.08417/) (2016)

## Abstract (arXiv)

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.