Inference for population genetics models is hindered by computationally intractable likelihoods. While this issue is tackled by likelihood-free methods, these approaches typically rely on hand-crafted summary statistics of the data. In complex settings, designing and selecting suitable summary statistics is problematic and results are very sensitive to such choices. In this paper, we learn the first exchangeable feature representation for population genetic data to work directly with genotype data. This is achieved by means of a novel Bayesian likelihood-free inference framework, where a permutation-invariant convolutional neural network learns the inverse functional relationship from the data to the posterior. We leverage access to scientific simulators to learn such likelihood-free function mappings, and establish a general framework for inference in a variety of simulation-based tasks. We demonstrate the power of our method on the recombination hotspot testing problem, outperforming the state-of-the-art.
@cite_24: 
@cite_29: A major goal of population genomics is to reconstruct the history of natural populations and to infer the neutral and selective scenarios that can explain the present-day polymorphism patterns. However, the separation between neutral and selective hypotheses has proven hard, mainly because both may predict similar patterns in the genome. This study focuses on the development of methods that can be used to distinguish neutral from selective hypotheses in equilibrium and nonequilibrium populations. These methods utilize a combination of statistics on the basis of the site frequency spectrum (SFS) and linkage disequilibrium (LD). We investigate the patterns of genetic variation along recombining chromosomes using a multitude of comparisons between neutral and selective hypotheses, such as selection or neutrality in equilibrium and nonequilibrium populations and recurrent selection models. We perform hypothesis testing using the classical P-value approach, but we also introduce methods from the machine-learning field. We demonstrate that the combination of SFS- and LD-based statistics increases the power to detect recent positive selection in populations that have experienced past demographic changes.
@cite_7: Given genomic variation data from multiple individuals, computing the likelihood of complex population genetic models is often infeasible. To circumvent this problem, we introduce a novel likelihood-free inference framework by applying deep learning, a powerful modern technique in machine learning. Deep learning makes use of multilayer neural networks to learn a feature-based function from the input (e.g., hundreds of correlated summary statistics of data) to the output (e.g., population genetic parameters of interest). We demonstrate that deep learning can be effectively employed for population genetic inference and learning informative features of data. As a concrete application, we focus on the challenging problem of jointly inferring natural selection and demography (in the form of a population size change history). Our method is able to separate the global nature of demography from the local nature of selection, without sequential steps for these two factors. Studying demography and selection jointly is motivated by Drosophila, where pervasive selection confounds demographic analysis. We apply our method to 197 African Drosophila melanogaster genomes from Zambia to infer both their overall demography, and regions of their genome under selection. We find many regions of the genome that have experienced hard sweeps, and fewer under selection on standing variation (soft sweep) or balancing selection. Interestingly, we find that soft sweeps and balancing selection occur more frequently closer to the centromere of each chromosome. In addition, our demographic inference suggests that previously estimated bottlenecks for African Drosophila melanogaster are too extreme.
@cite_20: Approximate Bayesian inference on the basis of summary statistics is well-suited to complex problems for which the likelihood is either mathematically or computationally intractable. However the methods that use rejection suffer from the curse of dimensionality when the number of summary statistics is increased. Here we propose a machine-learning approach to the estimation of the posterior density by introducing two innovations. The new method fits a nonlinear conditional heteroscedastic regression of the parameter on the summary statistics, and then adaptively improves estimation using importance sampling. The new algorithm is compared to the state-of-the-art approximate Bayesian methods, and achieves considerable reduction of the computational burden in two examples of inference in statistical genetics and in a queueing model.
Other likelihood-free approaches have emerged from the machine learning community and have been applied to population genetics, such as support vector machines (SVMs) @cite_29 @cite_24 , single-layer neural networks @cite_20 , and deep learning @cite_7 . The connection between likelihood-free Bayesian inference and neural networks has also been studied previously by and . An attractive property of these methods is that, unlike ABC, they can be applied to multiple datasets without repeating the training process, which is commonly referred to as amortized inference. However, current practice in population genetics collapses the data to a set of summary statistics before passing it through the machine learning models. Therefore, the performance still rests on the ability to laboriously hand-engineer informative statistics, and must be repeated from scratch for each new problem setting.