Highly tolerant likelihood - free Bayesian inference : An 1 adaptive non - linear heteroscedastic model

13 Approximate Bayesian inference on the basis of summary statistics is well14 suited to complex problems for which the likelihood is either mathematically 15 or computationally intractable. However the methods that use rejection suf16 fer from the curse of dimensionality when the number of summary statistics 17 is increased. Here we propose a machine-learning approach to the estimation 18 of the posterior density by introducing two innovations. The new method 19 fits a nonlinear conditional heteroscedastic regression of the parameter on 20 the summary statistics, and then adaptively improves estimation using im21 portance sampling. The new algorithm is compared to the state-of-the-art 22 approximate Bayesian methods, and achieves considerable reduction of the 23 computational burden in two examples of inference in statistical genetics and 24 in a queueing model. 25