  Bayesian nonparametric mixture models offer a rich framework for model based
clustering. We consider the situation where the kernel of the mixture is
available only up to an intractable normalizing constant. In this case, most of
the commonly used Markov chain Monte Carlo (MCMC) methods are not suitable. We
propose an approximate Bayesian computational (ABC) strategy, whereby we
approximate the posterior to avoid the intractability of the kernel. We derive
an ABC-MCMC algorithm which combines (i) the use of the predictive distribution
induced by the nonparametric prior as proposal and (ii) the use of the
Wasserstein distance and its connection to optimal matching problems. To
overcome the sensibility with respect to the parameters of our algorithm, we
further propose an adaptive strategy. We illustrate the use of the proposed
algorithm with several simulation studies and an application on real data,
where we cluster a population of networks, comparing its performance with
standard MCMC algorithms and validating the adaptive strategy.
