Optimally-Weighted Herding is Bayesian Quadrature

Details Optimally-Weighted Herding is Bayesian Quadrature Optimally-Weighted Herding is Bayesian Quadrature

2012-04-07

arxiv.org/abs/1204.1664v1

Statistics

Machine Learning

Scientific paper

Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is related to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves the optimal rate of convergence for any sampling method. We demonstrate empirically a rate of convergence faster than O(1/N). Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.

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