Interpolation error of misspecified Gaussian process regression

TL;DR

This paper analyzes how misspecification of covariance functions in Gaussian process regression affects interpolation error, showing that for certain kernels like Matern 1/2, parameter estimation errors have minimal impact, while for others it is more significant.

Contribution

It derives the interpolation error for misspecified Gaussian process models on infinite grids and compares the effects across different covariance functions.

Findings

Poor parameter estimation minimally impacts Matern 1/2 covariance interpolation.

Parameter misspecification significantly affects other covariance functions.

Numerical experiments validate theoretical insights.

Abstract

An interpolation error is an integral of the squared error of a regression model over a domain of interest. We consider the interpolation error for the case of misspecified Gaussian process regression: used covariance function differs from the true one. We derive the interpolation error for an infinite grid design of experiments. In particular, we show that for Matern 1/2 covariance function poor estimation of parameters only slightly affects the quality of interpolation. Then we proceed to numerical experiments that consider the misspecification for the most common covariance functions including other Matern and squared exponential covariance functions. For them, the quality of estimates of parameters affects the interpolation error.