Bayesian Kantorovich deconvolution in finite mixture models

This chapter addresses the problem of recovering the mixing distribution in finite kernel mixture models, when the number of components is unknown, yet bounded above by a fixed number. Taking a step back to the historical development of the analysis of this problem within the Bayesian paradigm and making use of the current methodology for the study of the posterior concentration phenomenon, we show that, for general prior laws supported over the space of mixing distributions with at most a fixed number of components, under replicated observations from the mixed density, the mixing distribution is estimable in the Kantorovich or $L^1$-Wasserstein metric at the optimal pointwise rate $n^{-1/4}$ (up to a logarithmic factor), $n$ being the sample size.