On the Sampling Problem for Kernel Quadrature
  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.
