Abstract
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, K., L. Song and A. Gretton (2010): “Kernel Bayes’ rule: Bayesian inference with positive definite kernels,” arXiv, 1009.5736 and Fukumizu, K., L. Song and A. Gretton (2011): “Kernel Bayes’ rule. Advances in Neural Information Processing Systems 24.” In: J. Shawe-Taylor and R. S. Zemel and P. Bartlett and F. Pereira and K. Q. Weinberger, (Eds.), pp. 1549–1557., 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.
We thank Mark Beaumont and Kevin Dawson for helpful discussions and comments. We also thank anonymous referees for helpful comments and the proposal of Example in the text. S.N. was supported in part by a Grant-in-Aid for the Japan Society for the Promotion of Science (JSPS) Research fellow (24-3234). K.F. was supported in part by JSPS KAKENHI (B) 22300098.
Appendix: proof of theorem
This proof is similar to the proof of Theorem 6.1 in Fukumizu et al. (2010). Denote variance and covariance operators by Css and Cθs, respectively, and the empirical estimators of them by
We want to establish
First we show
The left hand side is upper bounded by
By the decomposition
Therefore we have (3). Then, we show
The left hand side is upper bounded by
Where
Therefore we have (5). Then, (2) follows from (3) and (5). ■
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