Approximate Bayesian inference on the basis of summary statistics iswell-suited to complex problems for which the likelihood is eithermathematically or computationally intractable. However the methods that userejection suffer from the curse of dimensionality when the number of summarystatistics is increased. Here we propose a machine-learning approach to theestimation of the posterior density by introducing two innovations. The newmethod fits a nonlinear conditional heteroscedastic regression of the parameteron the summary statistics, and then adaptively improves estimation usingimportance sampling. The new algorithm is compared to the state-of-the-artapproximate Bayesian methods, and achieves considerable reduction of thecomputational burden in two examples of inference in statistical genetics andin a queueing model.
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