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., Matérn 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.

Classical quadrature rules via Gaussian processes

“…Properties of these rules were then studied in a number of articles during the 1970s (Richter, 1970;Richter-Dyn, 1971b,a;Larkin, 1972Larkin, , 1974Barrar et al, 1974;Barrar and Loeb, 1975;Bojanov, 1979). The early research was mostly concerned with optimal placement of the points of kernel cubature rules for totally positive kernels (Karlin, 1968) in one dimension; for more modern reviews of the topic, still incompletely understood, see Bojanov (1994), Karvonen et al (2019), and in particular Oettershagen (2017, Sec. 5.1).…”

“…The cubic computational cost in the number of integration points, due to the need to solve a linear system, is alleviated by Rathinavel and Hickernell (2018). Karvonen et al (2019) discuss qualitative properties, such as positivity, of the Bayesian cubature weights and Ehler et al (2019) are concerned with integration on general closed manifolds.…”

Adaptive Quadrature Schemes for Bayesian Inference via Active Learning

“…We could also employ different type of bases jointly, e.g., one different basis for each node. For instance, our framework allows the use of nearest neighbors (NN) basis functions, which presents several advantages: it does not require any matrix inversion and the coefficients of the linear combination (which defines the interpolator) are always positive [52], obtaining always a positive estimation of the marginal likelihood. These benefits are very appealing as shown in [14], [15], [52], [53].…”

A modern retrospective on probabilistic numerics

“…A notable feature of PN research since 2010 is the way that it has advanced on a broad front. The topic of quadrature/cubature, in the tradition of Sul din and Larkin, continues to be well represented: see, e.g., [Briol et al, 2019, Gunter et al, 2014, Karvonen et al, 2018b, Osborne et al, 2012a,b, Särkkä et al, 2016, Xi et al, 2018 and [Ehler et al, 2019, Jagadeeswaran and Hickernell, 2018, Karvonen et al, 2018a, 2019. The Bayesian approach to global optimisation continues to be widely used [Chen et al, 2018, Snoek et al, 2012, whilst probabilistic perspectives on quasi-Newton methods [Hennig and Kiefel, 2013] and line search methods [Mahsereci and Hennig, 2015] have been put forward.…”