Asymptotic analysis clarifies parameter estimates in gaussian processes
Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel
Machine Learning
Summary
Gaussian processes are mathematical tools used to make predictions based on data that change smoothly over space or time. A key part of these tools is the kernel, which helps define how points influence each other. The authors studied how well the method that guesses kernel settings from data—the maximum likelihood estimator—works when lots of closely spaced data are available. They showed exactly how these guesses get better with more data and proved they do so in the best possible way. This helps make Gaussian process models more trustworthy in fields like machine learning and statistics.
What this means in practice
- •For machine learning engineers: Use precise error rates to improve reliability when training Gaussian process models on dense data.
- •For spatial data analysts: Apply exact parameter behavior to enhance modeling and inference for spatially dense environmental measurements.
A theory result. No direct application yet.
Authors
Ameer Qaqish, Didong Li
Abstract
Gaussian processes (GPs) are widely used across machine learning, spatial statistics, time-series analysis, optimization, Bayesian statistics, and scientific applications. A central component of a GP model is its kernel, which is typically specified through a parametric family. Among the most widely used choices is the radial basis function (RBF), also known as the squared exponential or Gaussian kernel, owing to its simple form, smoothness, and flexibility. In practice, the kernel parameters are routinely estimated by the maximum likelihood estimators (MLEs), as implemented by standard GP software. Despite this widespread use, the asymptotic behavior of the MLEs remains poorly understood under fixed-domain asymptotics, even for the RBF kernel. The main difficulty arises from the increasingly strong dependence among densely sampled observations and the nonlinear dependence of the covariance matrix on the kernel parameters. In this paper, we address this gap by providing, to the best of our knowledge, the first complete asymptotic characterization of the joint MLE of the spatial variance, lengthscale, and nugget variance under fixed-domain asymptotics. We establish consistency, derive convergence rates for all three parameters, prove joint asymptotic normality, and show that these rates are minimax optimal.