Approximate Bayesian computation for finite mixture models

Finite mixture models are used in statistics and other disciplines, but inference for mixture models is challenging due, in part, to the multimodality of the likelihood function and the so-called label switching problem. We propose extensions of the Approximate Bayesian Computation-Population Monte Carlo (ABC-PMC) algorithm as an alternative framework for inference on finite mixture models. There are several decisions to make when implementing an ABC-PMC algorithm for finite mixture models, including the selection of the kernels used for moving the particles through the iterations, how to address the label switching problem and the choice of informative summary statistics. Examples are presented to demonstrate the performance of the proposed ABC-PMC algorithm for mixture modelling. The performance of the proposed method is evaluated in a simulation study and for the popular recessional velocity galaxy data.