On the positivity and magnitudes of Bayesian quadrature weights

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dc.contributorAalto-yliopistofi
dc.contributorAalto Universityen
dc.contributor.authorKarvonen, Tonien_US
dc.contributor.authorKanagawa, Motonobuen_US
dc.contributor.authorSärkkä, Simoen_US
dc.contributor.departmentDepartment of Electrical Engineering and Automationen
dc.contributor.groupauthorSensor Informatics and Medical Technologyen
dc.contributor.organizationUniversity of Tübingenen_US
dc.date.accessioned2019-11-07T12:02:20Z
dc.date.available2019-11-07T12:02:20Z
dc.date.issued2019-10-04en_US
dc.description.abstractThis 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.en
dc.description.versionPeer revieweden
dc.format.extent17
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationKarvonen, T, Kanagawa, M & Särkkä, S 2019, ' On the positivity and magnitudes of Bayesian quadrature weights ', STATISTICS AND COMPUTING . https://doi.org/10.1007/s11222-019-09901-0en
dc.identifier.doi10.1007/s11222-019-09901-0en_US
dc.identifier.issn0960-3174
dc.identifier.otherPURE UUID: 2f899f18-8100-4a71-a911-3206f743a3a9en_US
dc.identifier.otherPURE ITEMURL: https://research.aalto.fi/en/publications/2f899f18-8100-4a71-a911-3206f743a3a9en_US
dc.identifier.otherPURE FILEURL: https://research.aalto.fi/files/38302659/ELEC_Karvonen_etal_On_the_Posivitity_and_Magnitudes_StatandComp_2019_finalpublishedversion.pdfen_US
dc.identifier.urihttps://aaltodoc.aalto.fi/handle/123456789/41047
dc.identifier.urnURN:NBN:fi:aalto-201911076052
dc.language.isoenen
dc.publisherSPRINGER
dc.relation.ispartofseriesSTATISTICS AND COMPUTINGen
dc.rightsopenAccessen
dc.subject.keywordBayesian quadratureen_US
dc.subject.keywordProbabilistic numericsen_US
dc.subject.keywordGaussian processesen_US
dc.subject.keywordChebyshev systemsen_US
dc.subject.keywordStabilityen_US
dc.subject.keywordHilbert-spacesen_US
dc.subject.keywordApproximationen_US
dc.subject.keywordInterpolationen_US
dc.subject.keywordCubatureen_US
dc.subject.keywordFormulasen_US
dc.titleOn the positivity and magnitudes of Bayesian quadrature weightsen
dc.typeA1 Alkuperäisartikkeli tieteellisessä aikakauslehdessäfi
dc.type.versionpublishedVersion
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