  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.
