Summary: Genetic data obtained on population samples convey information about their evolutionary history. Inference methods can extract part of this information but they require sophisticated statistical techniques that have been made available to the biologist community (through computer programs) only for simple and standard situations typically involving a small number of samples. We propose here a computer program (DIY ABC) for inference based on approximate Bayesian computation (ABC), in which scenarios can be customized by the user to fit many complex situations involving any number of populations and samples. Such scenarios involve any combination of population divergences, admixtures and population size changes. DIY ABC can be used to compare competing scenarios, estimate parameters for one or more scenarios and compute bias and precision measures for a given scenario and known values of parameters (the current version applies to unlinked microsatellite data). This article describes key methods used in the program and provides its main features. The analysis of one simulated and one real dataset, both with complex evolutionary scenarios, illustrates the main possibilities of DIY ABC.Availability: The software DIY ABC is freely available at http://www.montpellier.inra.fr/CBGP/diyabc.Contact: j.cornuet@imperial.ac.ukSupplementary information: Supplementary data are also available at http://www.montpellier.inra.fr/CBGP/diyabc

Also known as likelihood-free methods, approximate Bayesian computational (ABC) methods have appeared in the past ten years as the most satisfactory approach to intractable likelihood problems, first in genetics then in a broader spectrum of applications. However, these methods suffer to some degree from calibration difficulties that make them rather volatile in their implementation and thus render them suspicious to the users of more traditional Monte Carlo methods. In this survey, we study the various improvements and extensions brought on the original ABC algorithm in recent years.

Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappé et al. (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm.

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In this paper, we propose an adaptive algorithm that iteratively updates both the weights and component parameters of a mixture importance sampling density so as to optimise the performance of importance sampling, as measured by an entropy criterion. The method, called M-PMC, is shown to be applicable to a wide class of importance sampling densities, which includes in particular mixtures of multivariate Student t distributions. The performance of the proposed scheme is studied on both artificial and real examples, highlighting in particular the benefit of a novel Rao-Blackwellisation device which can be easily incorporated in the updating scheme.

'But, as you have already pointed out, we do not need any more disjointed clues,' said Bartholomew. 'That has been our problem all along: we have a mass of small facts and small scraps of information, but we are unable to make any sense out of them. The last thing we need is more.' Susanna Gregory, A Summer of Discontent IntroductionToday's data analysts and modellers are in the luxurious position of being able to more closely describe, estimate, predict and infer about complex systems of interest, thanks to ever more powerful computational methods but also wider ranges of modelling distributions. Mixture models constitute a fascinating illustration of these aspects: while within a parametric family, they offer malleable approximations in non-parametric settings; although based on standard distributions, they pose highly complex computational challenges; and they are both easy to constrain to meet identifiability requirements and fall within the class of ill-posed problems. They also provide an endless benchmark for assessing new techniques, from the EM algorithm to reversible jump methodology. In particular, they exemplify the

Approximate Bayesian computation (ABC) have become an essential tool for the analysis of complex stochastic models. Grelaud et al. [(2009) Bayesian Anal 3:427-442] advocated the use of ABC for model choice in the specific case of Gibbs random fields, relying on an intermodel sufficiency property to show that the approximation was legitimate. We implemented ABC model choice in a wide range of phylogenetic models in the Do It Yourself-ABC (DIY-ABC) software [Cornuet et al. (2008) Bioinformatics 24:2713-2719. We now present arguments as to why the theoretical arguments for ABC model choice are missing, because the algorithm involves an unknown loss of information induced by the use of insufficient summary statistics. The approximation error of the posterior probabilities of the models under comparison may thus be unrelated with the computational effort spent in running an ABC algorithm. We then conclude that additional empirical verifications of the performances of the ABC procedure as those available in DIY-ABC are necessary to conduct model choice.Bayes factor | Bayesian model choice | likelihood-free methods | sufficient statistics | consistent tests I nference on population genetic models such as coalescent trees is one representative example of cases when statistical analyses such as Bayesian inference cannot easily operate because the likelihood function associated with the data cannot be computed in a manageable time (1-3). The fundamental reason for this impossibility is that the model associated with coalescent data has to integrate over trees of high complexity.In such settings, traditional approximation tools such as Monte Carlo simulation (4) from the posterior distribution are unavailable for practical purposes. Indeed, due to the complexity of the latent structures defining the likelihood (like the coalescent tree), their simulation is too unstable to bring a reliable approximation in a manageable time. Such complex models call for a practical if cruder approximation method, the approximate Bayesian computation (ABC) methodology (1, 5). This rejection technique bypasses the computation of the likelihood via simulations from the corresponding distribution (see refs. 6 and 7 for recent surveys, and ref. 8 for the wide and successful array of applications based on implementations of ABC in genomics and ecology).We argue here that ABC is a generally valid approximation method for doing Bayesian inference in complex models. However, without further justification, ABC methods cannot be trusted to discriminate between two competing models when based on insufficient summary statistics. We exhibit simple examples in which the information loss due to insufficiency leads to inconsistency, i.e., when the ABC model selection fails to recover the true model, even with infinite amounts of observation and computation. On the one hand, ABC using the entire data leads to a consistent model-choice decision, but it is clearly infeasible in most settings. On the other hand, too much information loss due to insufficiency l...

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