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Validating approximate Bayesian computation on posterior convergence

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SINW01 - Scalable statistical inference

Co-author: Paul Fearnhead (Lancaster University)

Many statistical applications involve models for which it is difficult to evaluate the likelihood, but relatively easy to sample from. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present a number of surprisingly strong asymptotic results for the regression-adjusted version of approximate Bayesian Computation introduced by Beaumont et al. (2002). We show that for an appropriate choice of the bandwidth in approximate Bayesian computation, using regression-adjustment will lead to a posterior that, asymptotically, correctly quantifies uncertainty. Furthermore, for such a choice of bandwidth we can implement an importance sampling algorithm to sample from the posterior whose acceptance probability tends to 1 as we increase the data sample size. This compares favourably to results for standard approximate Bayesian computation, where the only way to obtain its posterior that correctly quantifies uncertainty is to choose a much smaller bandwidth, for which the acceptance probability tends to 0 and hence for which Monte Carlo error will dominate.

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