Statistical approaches in population genetics have two distinct objectives, which consist of describing the data and of inferring the evolutionary processes that generated the observed patterns. The first chapter of this thesis describes my contributions to Approximate Bayesian Computation (ABC), which allows to compare and to infer the evolutionary processes that shaped genetic variation. First, I describe asymptotic results, which provide biases and variances of posterior estimates obtained with approximate Bayesian Computation. The results highlight what are the benefits of using regression-adjustment methods and of reducing the dimension of the descriptive statistics used in ABC. Then, I present an original method for ABC that both performs regression-adjustment and dimension reduction. An analysis where we compare different methods of dimension reduction ends the first chapter. The second chapter of the thesis is devoted to the goal of describing the data in a spatial context. The statistical methods we propose are based on the concept of isolation by distance (IBD), which is a particular form of spatial autocorrelation where correlation decays with distance. With a Kriging approach, we can characterize non-stationary patterns of isolation by distance where the decay of correlation with distance varies over the sampling range. We also propose an anisotropic extension of the concept of isolation by distance and we provide a characterization and a test for anisotropy using a regression equation. The conclusion of this thesis deals with some important caveats: the difficulty of interpreting statistical results, the robustness of the results with respect to the sampling scheme and the too often neglected goodness-of-fit. The thesis ends with some perspectives about how Bayesian methods could scale with the massive dimension of the data produced in genetics.
Bayesian statistics and applications to populations genetics