Abstract
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 equivalent 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 performance superior to any other weighted herding 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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