This document is an invited chapter covering the specificities of ABC model choice, intended for the incoming Handbook of ABC by Sisson, Fan, and Beaumont (2017). Beyond exposing the potential pitfalls of ABC based posterior probabilities, the review
emphasizes mostly the solution proposed by Pudlo et al. (2016) on the use of random forests for aggregating summary statistics and and for estimating the posterior probability of the most likely model via a secondary random fores.
Approximate Bayesian computation (ABC) methods provide an elaborate approach to Bayesian inference on complex models, including model choice. Both theoretical arguments and simulation experiments indicate, however, that model posterior probabilities
may be poorly evaluated by standard ABC techniques. We propose a novel approach based on a machine learning tool named random forests to conduct selection among the highly complex models covered by ABC algorithms. We thus modify the way Bayesian model selection is both understood and operated, in that we rephrase the inferential goal as a classification problem, first predicting the model that best fits the data with random forests and postponing the approximation of the posterior probability of the predicted MAP for a second stage also relying on random forests. Compared with earlier implementations of ABC model choice, the ABC random forest approach offers several potential improvements: (i) it often has a larger discriminative power among the competing models, (ii) it is more robust against the number and choice of statistics summarizing the data, (iii) the computing effort is drastically reduced (with a gain in computation efficiency of at least fifty), and (iv) it includes an approximation of the posterior probability of the selected model. The call to random forests will undoubtedly extend the range of size of datasets and complexity of models that ABC can handle. We illustrate the power of this novel methodology by analyzing controlled experiments as well as genuine population genetics datasets. The proposed methodologies are implemented in the R package abcrf available on the CRAN.
Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. The Adaptive Multiple Importance Sampling (AMIS) of Cornuet et al. (2012) provides a sign
ificant improvement in stability and effective sample size due to the introduction of a recycling procedure. However, the consistency of the AMIS estimator remains largely open. In this work we prove the convergence of the AMIS, at a cost of a slight modification in the learning process. Contrary to Douc et al. (2007a), results are obtained here in the asymptotic regime where the number of iterations is going to infinity while the number of drawings per iteration is a fixed, but growing sequence of integers. Hence some of the results shed new light on adaptive population Monte Carlo algorithms in that last regime.
