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.

翻译：本文批判性地审视了在单输出高斯过程模型中普遍采用马特恩核加性混合的做法，并探讨了多输出高斯过程模型中马特恩核乘性混合的特性。针对单输出情形，我们推导了一系列理论结果，表明马特恩核混合的光滑性由最不光滑的组分决定，且具有此类核的高斯过程实际上等价于最不光滑的核组分。此外，我们证明了混合权重或各核组分内部的参数均不可辨识。随后，我们聚焦多输出高斯过程模型，分析了乘性核 $K(x,y) = AK_0(x,y)$ 中协方差矩阵 $A$ 的可辨识性，其中 $K_0$ 为标准单输出核（如马特恩核）。研究表明，$A$ 可辨识至多乘性常数，这表明乘性混合非常适合多输出任务。我们的发现得到了针对单输出与多输出场景的广泛模拟及实际应用的支持。本工作为高斯过程模型的核选择与解释提供了洞见，强调了针对不同任务选择适当核结构的重要性。