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The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio s/d, where s and d encode the smoothness and dimension of the integrand. However, an empirical investigation reveals that the rate constant C is highly sensitive to the distribution of the random points. In contrast to standard Monte Carlo integration, for which optimal importance sampling is well-understood, the sampling distribution that minimises C for Kernel Quadrature does not admit a closed form. This paper argues that the practical choice of sampling distribution is an important open problem. One solution is considered; a novel automatic approach based on adaptive tempering and sequential Monte Carlo. Empirical results demonstrate a dramatic reduction in integration error of up to 4 orders of magnitude can be achieved with the proposed method.
Author Information
Francois-Xavier Briol (University of Warwick)
Chris J Oates (Newcastle University)
Jon Cockayne (University of Warwick)
Wilson Ye Chen (University of Technology Sydney)
Mark Girolami (Imperial College London)
Related Events (a corresponding poster, oral, or spotlight)
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2017 Poster: On the Sampling Problem for Kernel Quadrature »
Tue. Aug 8th 08:30 AM -- 12:00 PM Room Gallery #94
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