On the Identifiability and Interpretability of Gaussian Process Models

TL;DR

This paper investigates the properties of additive and multiplicative Matérn kernels in Gaussian process models, revealing limitations in identifiability and interpretability for single-output models and advantages for multi-output models.

Contribution

It provides theoretical insights into the smoothness and parameter identifiability of additive and multiplicative Matérn kernels in Gaussian processes, guiding better kernel selection.

Findings

Single-output kernel smoothness is determined by the least smooth component.

Kernel parameters in additive mixtures are not identifiable.

In multi-output models, the covariance matrix is identifiable up to a constant.

Abstract

In this paper, we critically examine the prevalent practice of using additive mixtures of Mat\'ern kernels in single-output Gaussian process (GP) models and explore the properties of multiplicative mixtures of Mat\'ern kernels for multi-output GP models. For the single-output case, we derive a series of theoretical results showing that the smoothness of a mixture of Mat\'ern kernels is determined by the least smooth component and that a GP with such a kernel is effectively equivalent to the least smooth kernel component. Furthermore, we demonstrate that none of the mixing weights or parameters within individual kernel components are identifiable. We then turn our attention to multi-output GP models and analyze the identifiability of the covariance matrix $A$ in the multiplicative kernel $K(x,y) = AK_0(x,y)$, where $K_0$ is a standard single output kernel such as Mat\'ern. We show that $A$ is identifiable up to a multiplicative constant, suggesting that multiplicative mixtures are well suited for multi-output tasks. Our findings are supported by extensive simulations and real applications for both single- and multi-output settings. This work provides insight into kernel selection and interpretation for GP models, emphasizing the importance of choosing appropriate kernel structures for different tasks.