Approximate Bayesian Computation in Population Genetics

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Abstract
We propose a new method for approximate Bayesian statistical inference on the basis of summary statistics. The method is suited to complex problems that arise in population genetics, extending ideas developed in this setting by earlier authors. Properties of the posterior distribution of a parameter, such as its mean or density curve, are approximated without explicit likelihood calculations. This is achieved by fitting a local-linear regression of simulated parameter values on simulated summary statistics, and then substituting the observed summary statistics into the regression equation. The method combines many of the advantages of Bayesian statistical inference with the computational efficiency of methods based on summary statistics. A key advantage of the method is that the nuisance parameters are automatically integrated out in the simulation step, so that the large numbers of nuisance parameters that arise in population genetics problems can be handled without difficulty. Simulation results indicate computational and statistical efficiency that compares favorably with those of alternative methods previously proposed in the literature. We also compare the relative efficiency of inferences obtained using methods based on summary statistics with those obtained directly from the data using MCMC.