On the positivity and magnitudes of Bayesian quadrature weights

Toni Karvonen*, Motonobu Kanagawa, Simo Särkkä

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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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. Matern 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.

Original languageEnglish
Number of pages17
Publication statusPublished - 4 Oct 2019
MoE publication typeA1 Journal article-refereed


  • Bayesian quadrature
  • Probabilistic numerics
  • Gaussian processes
  • Chebyshev systems
  • Stability
  • Hilbert-spaces
  • Approximation
  • Interpolation
  • Cubature
  • Formulas


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