A probabilistic Taylor expansion with Gaussian processes
This work provides a theoretical framework for Gaussian process regression with derivative data, which is incremental in the field of probabilistic modeling and kernel methods.
The authors tackled the problem of replicating truncated Taylor expansions using Gaussian processes by introducing Taylor kernels and a specific data configuration of derivative evaluations, achieving exact replication of any order Taylor expansion as the posterior mean.
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.