On the positivity and magnitudes of Bayesian quadrature weights

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On the positivity and magnitudes of Bayesian quadrature weights. / Karvonen, Toni; Kanagawa, Motonobu; Särkkä, Simo.

julkaisussa: STATISTICS AND COMPUTING, 04.10.2019.

Tutkimustuotos: Lehtiartikkelivertaisarvioitu

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Bibtex - Lataa

@article{2f899f1881004a71a9113206f743a3a9,
title = "On the positivity and magnitudes of Bayesian quadrature weights",
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.",
keywords = "Bayesian quadrature, Probabilistic numerics, Gaussian processes, Chebyshev systems, Stability, Hilbert-spaces, Approximation, Interpolation, Cubature, Formulas",
author = "Toni Karvonen and Motonobu Kanagawa and Simo S{\"a}rkk{\"a}",
year = "2019",
month = "10",
day = "4",
doi = "10.1007/s11222-019-09901-0",
language = "English",
journal = "STATISTICS AND COMPUTING",
issn = "0960-3174",

}

RIS - Lataa

TY - JOUR

T1 - On the positivity and magnitudes of Bayesian quadrature weights

AU - Karvonen, Toni

AU - Kanagawa, Motonobu

AU - Särkkä, Simo

PY - 2019/10/4

Y1 - 2019/10/4

N2 - 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.

AB - 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.

KW - Bayesian quadrature

KW - Probabilistic numerics

KW - Gaussian processes

KW - Chebyshev systems

KW - Stability

KW - Hilbert-spaces

KW - Approximation

KW - Interpolation

KW - Cubature

KW - Formulas

U2 - 10.1007/s11222-019-09901-0

DO - 10.1007/s11222-019-09901-0

M3 - Article

JO - STATISTICS AND COMPUTING

JF - STATISTICS AND COMPUTING

SN - 0960-3174

ER -

ID: 38038034