# On the positivity and magnitudes of Bayesian quadrature weights

## TL;DR

This paper analyzes the properties of Bayesian quadrature weights, focusing on conditions for positivity and bounds on their magnitudes, to improve stability and robustness of quadrature rules.

## Contribution

It provides theoretical conditions for positive weights in univariate cases and bounds on weight magnitudes based on point distribution and kernel properties.

## Key findings

- Weights are positive if points minimize posterior variance and kernel is totally positive.
- Weight magnitudes are bounded by fill distance and separation radius in Sobolev spaces.
- Numerical examples suggest potential for further generalizations and improvements.

## Abstract

This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g., Mat\'ern kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.