# Marginalising over Stationary Kernels with Bayesian Quadrature

arXiv: https://arxiv.org/abs/2106.07452 · HTML: https://mlsanity.com/paper/2106.07452/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 56 references that are in the map.
- First posted on arXiv 14 Jun 2021.

## Builds on (most cited first, 26 of 26)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [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)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Preconditioning Kernel Matrices](https://mlsanity.com/paper/1602.06693/) (2016)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)
- [Fast Information-theoretic Bayesian Optimisation](https://mlsanity.com/paper/1711.00673/) (2017)
- [Generalized Spectral Kernels](https://mlsanity.com/paper/1506.02236/) (2015)
- [Active Multi-Information Source Bayesian Quadrature](https://mlsanity.com/paper/1903.11331/) (2019)
- [Function-Space Distributions over Kernels](https://mlsanity.com/paper/1910.13565/) (2019)
- [Bayesian Quadrature for Multiple Related Integrals](https://mlsanity.com/paper/1801.04153/) (2018)
- [A Gaussian Process Regression Model for Distribution Inputs](https://mlsanity.com/paper/1701.09055/) (2017)
- [Improving Quadrature for Constrained Integrands](https://mlsanity.com/paper/1802.04782/) (2018)
- [Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs](https://mlsanity.com/paper/1503.02424/) (2015)
- [Bayesian Nonparametric Kernel-Learning](https://mlsanity.com/paper/1506.08776/) (2015)
- [Scalable L'evy Process Priors for Spectral Kernel Learning](https://mlsanity.com/paper/1802.00530/) (2018)
- [Bayesian Nonparametric Spectral Estimation](https://mlsanity.com/paper/1809.02196/) (2018)
- [Scalable Variational Bayesian Kernel Selection for Sparse Gaussian Process Regression](https://mlsanity.com/paper/1912.02641/) (2019)
- [Automated Model Selection with Bayesian Quadrature](https://mlsanity.com/paper/1902.09724/) (2019)
- [Integral Transforms from Finite Data: An Application of Gaussian Process Regression to Fourier Analysis](https://mlsanity.com/paper/1704.02828/) (2017)

## Built on by (most cited first, 4 of 4)

- [Fast Bayesian Inference with Batch Bayesian Quadrature via Kernel Recombination](https://mlsanity.com/paper/2206.04734/) (2022)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Stein Random Feature Regression](https://mlsanity.com/paper/2406.00438/) (2024)

## Abstract (arXiv)

Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive for larger datasets. We propose a Bayesian Quadrature scheme to make this marginalisation more efficient and thereby more practical. Through use of the maximum mean discrepancies between distributions, we define a kernel over kernels that captures invariances between Spectral Mixture (SM) Kernels. Kernel samples are selected by generalising an information-theoretic acquisition function for warped Bayesian Quadrature. We show that our framework achieves more accurate predictions with better calibrated uncertainty than state-of-the-art baselines, especially when given limited (wall-clock) time budgets.