# On the Identifiability and Interpretability of Gaussian Process Models

arXiv: https://arxiv.org/abs/2310.17023 · HTML: https://mlsanity.com/paper/2310.17023/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 25 Oct 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Structure Discovery in Nonparametric Regression through Compositional Kernel Search](https://mlsanity.com/paper/1302.4922/) (2013)
- [Automatic Construction and Natural-Language Description of Nonparametric Regression Models](https://mlsanity.com/paper/1402.4304/) (2014)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Differentiable Compositional Kernel Learning for Gaussian Processes](https://mlsanity.com/paper/1806.04326/) (2018)
- [Generalized Spectral Kernels](https://mlsanity.com/paper/1506.02236/) (2015)
- [Kernel Identification Through Transformers](https://mlsanity.com/paper/2106.08185/) (2021)
- [Cross-Covariance Functions for Multivariate Geostatistics](https://mlsanity.com/paper/1507.08017/) (2015)
- [Evolution of Covariance Functions for Gaussian Process Regression using Genetic Programming](https://mlsanity.com/paper/1305.3794/) (2013)

## Built on by (most cited first, 2 of 2)

- [A class of modular and flexible covariate-based covariance functions for nonstationary spatial modeling](https://mlsanity.com/paper/2410.16716/) (2024)
- [Data-efficient Bayesian-guided design selection from large candidate sets: Application to hyperelastic stochastic metamaterials](https://mlsanity.com/paper/2603.15917/) (2026)

## Abstract (arXiv)

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) = AK0(x,y), where K0 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.