  Approximate Bayesian computation (ABC) is a likelihood-free approach for
Bayesian inferences based on a rejection algorithm method that applies a
tolerance of dissimilarity between summary statistics from observed and
simulated data. Although several improvements to the algorithm have been
proposed, none of these improvements avoid the following two sources of
approximation: 1) lack of sufficient statistics: sampling is not from the true
posterior density given data but from an approximate posterior density given
summary statistics; and 2) non-zero tolerance: sampling from the posterior
density given summary statistics is achieved only in the limit of zero
tolerance. The first source of approximation can be improved by adding a
summary statistic, but an increase in the number of summary statistics could
introduce additional variance caused by the low acceptance rate. Consequently,
many researchers have attempted to develop techniques to choose informative
summary statistics. The present study evaluated the utility of a kernel-based
ABC method (Fukumizu et al. 2010, arXiv:1009.5736 and 2011, NIPS 24: 1549-1557)
for complex problems that demand many summary statistics. Specifically, kernel
ABC was applied to population genetic inference. We demonstrate that, in
contrast to conventional ABCs, kernel ABC can incorporate a large number of
summary statistics while maintaining high performance of the inference.
