Adaptive Importance Sampling and ABC methods with application to population genetics
This thesis consists of two parts which can be read independently. The first part is about the Adaptive Multiple Importance Sampling (AMIS) algorithm presented in Cornuet et al. provides a significant improvement in stability and Effective Sample Size due to the introduction of the recycling procedure. These numerical properties are particularly adapted to the Bayesian paradigm in population genetics where the modelization involves a large number of parameters. However, the consistency of the AMIS estimator remains largely open. In this work, we provide a novel Adaptive Multiple Importance Sampling scheme corresponding to a slight modification of Cornuet et al. proposition that preserves the above-mentioned improvements. Finally, using limit theorems on triangular arrays of conditionally independant random variables, we give a consistensy result for the final particle system returned by our new scheme. The second part of this thesis lies in ABC paradigm. Approximate Bayesian Computation has been successfully used in population genetics models to bypass the calculation of the likelihood. These algorithms provide an accurate estimator by comparing the observed dataset to a sample of datasets simulated from the model. Although parallelization is easily achieved, computation times for assuring a suitable approximation quality of the posterior distribution are still long. To alleviate this issue, we propose a sequential algorithm adapted from Del Moral et al. which runs twice as fast as traditional ABC algorithms. Its parameters are calibrated to minimize the number of simulations from the model.