Fast automatic Bayesian cubature using lattice sampling

Jagadeeswaran, R.; Hickernell, Fred J.

doi:10.1007/s11222-019-09895-9

Cited by 25 publications

(19 citation statements)

References 24 publications

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“…For example, in a quadrature setting, the practitioner is in the fortunate position of being able to use evaluations of the integrand u both to estimate the regularity of u and the value of the integral. Empirical Bayesian methods are explored in [Schober et al, 2018] and in [Jagadeeswaran and Hickernell, 2018].…”

Section: Adaptive Bayesian Methodsmentioning

confidence: 99%

A modern retrospective on probabilistic numerics

Oates

Sullivan

2019

Stat Comput

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This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.

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Section: Adaptive Bayesian Methodsmentioning

confidence: 99%

A modern retrospective on probabilistic numerics

Oates

Sullivan

2019

Stat Comput

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“…As pointed out by Owen, Hickernell and Jagadeeswaran, this may still usually require O(n 3 ) operations, but many approximation schemes for Gaussian processes exist. Some cheaper exact schemes based on particular choices of point sets or covariance functions have also recently been developed for the specific case of Bayesian numerical methods Karvonen and Särkkä [2018], Jagadeeswaran and Hickernell [2018]. Furthermore, we believe that the approach based on Fourier transforms suggested by Stein and Hung could also be fruitful.…”

Section: Bayesian Priors Posteriors and The Difficulty Of Working With Measuresmentioning

confidence: 99%

Rejoinder: Probabilistic Integration: A Role in Statistical Computation?

Briol¹,

Oates²,

Girolami³

et al. 2019

Statist. Sci.

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This article is the rejoinder for the paper "Probabilistic Integration: A Role in Statistical Computation?" to appear in Statistical Science with discussion [Briol et al., 2015]. We would first like to thank the reviewers and many of our colleagues who helped shape this paper, the editor for selecting our paper for discussion, and of course all of the discussants for their thoughtful, insightful and constructive comments. In this rejoinder, we respond to some of the points raised by the discussants and comment further on the fundamental questions underlying the paper:• Should Bayesian ideas be used in numerical analysis?• If so, what role should such approaches have in statistical computation? Bayesian Numerical MethodsNumerical analysis is concerned with the approximation of typically high or infinite-dimensional mathematical quantities using discretisations of the space on which these are defined. Examples include integrands, and as a consequence their corresponding integrals, or the solutions of differential equations. Different discretisation schemes lead to different numerical algorithms, whose stability and convergence properties need to be carefully assessed. Such numerical methods have undeniably played, and continue to play, an important role in the implementation of statistical methods, from the computation of intractable integrals in Bayesian statistics to the computation of estimators using optimisation routines in frequentist statistics.

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“…-There is convincing numerical evidence that the weights are positive if the nodes for the Gaussian kernel and measure on R are selected by suitable scaling the classical Gauss-Hermite nodes [35]. -Uniform weighting (i.e., w BQ X,i = 1/n) can be achieved when certain quasi-Monte Carlo point sets and shift invariant kernels are used [27].…”

Section: Other Kernels and Point Setsmentioning

confidence: 99%

On the positivity and magnitudes of Bayesian quadrature weights

2019

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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., 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.

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Fast automatic Bayesian cubature using lattice sampling

Cited by 25 publications

References 24 publications

A modern retrospective on probabilistic numerics

A modern retrospective on probabilistic numerics

Rejoinder: Probabilistic Integration: A Role in Statistical Computation?

On the positivity and magnitudes of Bayesian quadrature weights

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