On the positivity and magnitudes of Bayesian quadrature weights
Loading...
Access rights
openAccess
publishedVersion
URL
Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
This publication is imported from Aalto University research portal.
View publication in the Research portal (opens in new window)
View/Open full text file from the Research portal (opens in new window)
View publication in the Research portal (opens in new window)
View/Open full text file from the Research portal (opens in new window)
Date
Major/Subject
Mcode
Degree programme
Language
en
Pages
17
Series
STATISTICS AND COMPUTING
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. 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.Description
Other note
Citation
Karvonen, T, Kanagawa, M & Särkkä, S 2019, 'On the positivity and magnitudes of Bayesian quadrature weights', STATISTICS AND COMPUTING. https://doi.org/10.1007/s11222-019-09901-0