R2-GP: Variance-Calibrated Priors for Additive Gaussian Processes
Abstract
We develop R2-GP, an R2D2-type prior for Gaussian process (GP) generalised additive models (GAMs) that calibrates explained variance (R2) for Matérn covariance kernels. The function variance for standard Matérn kernels depends jointly on the effect magnitude and length-scale. As such, smooth and rough components with identical amplitudes contribute very different amounts of variation. With independent priors on component amplitudes, this variation results in variance overestimation, especially in higher dimensions. We overcome this by taking direct control over each component's expected contribution to prior explained variance. We do this by rescaling each component based on its length scale and observed inputs, and modelling each rescaled variance as a share of the function variance induced by a global R2 prior. This allows R2-GP to scale well to many components. To also scale to large sample sizes, we extend R2-GP to Hilbert space approximation methods, a framework for kernels exploiting their spectral decompositions for fast and efficient inference. Our simulations and real-world case studies demonstrate superior generalisation and parameter recovery over independent variance priors.
Related
- Hypothesis Testing for Nonlinear and Interactive Effects Using Additive Gaussian Processes
- Higher Order Risk Comparisons of Oracle Calibrated Global Local Priors
- Proximal R2 Priors for Structured Variance Allocation
- A tutorial on selecting, translating, and reporting variance priors in Bayesian multilevel models.
- Breaking the Likelihood Trap: Variance-Calibrated Modulation for Large Language Model Decoding
BibTeX
@misc{aguilar2026r2gp,
title = {{R2-GP: Variance-Calibrated Priors for Additive Gaussian Processes}},
author = {Javier Enrique Aguilar and Soham Mukherjee and David Kohns},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26437},
eprint = {2610.26437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26437},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - R2-GP: Variance-Calibrated Priors for Additive Gaussian Processes
AU - Javier Enrique Aguilar
AU - Soham Mukherjee
AU - David Kohns
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26437
UR - https://arxiv.org/abs/2610.26437
AB - We develop R2-GP, an R2D2-type prior for Gaussian process (GP) generalised additive models (GAMs) that calibrates explained variance (R^2) for Matérn covariance kernels. The function variance for standard Matérn kernels depends jointly on the effect magnitude and length-scale. As such, smooth and rough components with identical amplitudes contribute very different amounts of variation. With independent priors on component amplitudes, this variation results in variance overestimation, especially in higher dimensions. We overcome this by taking direct control over each component's expected contribution to prior explained variance. We do this by rescaling each component based on its length scale and observed inputs, and modelling each rescaled variance as a share of the function variance induced by a global R^2 prior. This allows R2-GP to scale well to many components. To also scale to large sample sizes, we extend R2-GP to Hilbert space approximation methods, a framework for kernels exploiting their spectral decompositions for fast and efficient inference. Our simulations and real-world case studies demonstrate superior generalisation and parameter recovery over independent variance priors.
ER -