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Within a Bayesian framework, a comprehensive investigation of mixtures offinite mixtures (MFMs), i.e., finite mixtures with a prior on the number ofcomponents, is performed. This model class has applications in model-basedclustering as well as for semi-parametric density estimation and requiressuitable prior specifications and inference methods to exploit its fullpotential. We contribute by considering a generalized class of MFMs where thehyperparameter $\gamma_K$ of a symmetric Dirichlet prior on the weightdistribution depends on the number of components. We show that this model classmay be regarded as a Bayesian non-parametric mixture outside the class ofGibbs-type priors. We emphasize the distinction between the number ofcomponents $K$ of a mixture and the number of clusters $K_+$, i.e., the numberof filled components given the data. In the MFM model, $K_+$ is a randomvariable and its prior depends on the prior on $K$ and on the hyperparameter$\gamma_K$. We employ a flexible prior distribution for the number ofcomponents $K$ and derive the corresponding prior on the number of clusters$K_+$ for generalized MFMs. For posterior inference, we propose the noveltelescoping sampler which allows Bayesian inference for mixtures with arbitrarycomponent distributions without resorting to reversible jump Markov chain MonteCarlo (MCMC) methods. The telescoping sampler explicitly samples the number ofcomponents, but otherwise requires only the usual MCMC steps of a finitemixture model. The ease of its application using different componentdistributions is demonstrated on several data sets.

Generalized mixtures of finite mixtures and telescoping sampling