Hierarchical mixture modelling with normalized inverse Gaussian priors

In recent years the Dirichlet process prior has experienced a great success in the context of Bayesian mixture modelling. The idea of overcoming discreteness of its realizations by exploiting it in hierarchical models, combined with the development of suitable sampling techniques, represent one of the reasons of its popularity. In this paper we aim at proposing the normalized inverse Gaussian process as an alternative to the Dirichlet process to be used in Bayesian hierarchical models. The normalized inverse Gaussian prior is constructed via its finite-dimensional distributions. This prior, though sharing the discreteness property of the Dirichlet prior, is characterized by a more elaborate and sensible clustering which makes use of all the information contained in the data. While in the Dirichlet case the mass assigned to each observation depends solely on the number of times it occurred, for the normalized inverse Gaussian prior the weight of a single observation heavily depends on the whole number of ties in the sample. Moreover, expressions corresponding to relevant statistical quantities, such as a priori moments and the predictive distributions, are as tractable as those arising from the Dirichlet process. This implies that well-established sampling schemes can be easily extended to cover hierarchical models based upon the normalized inverse Gaussian process. The mixture of normalized inverse Gaussian process and the mixture of Dirichlet process are compared by means of two examples involving mixtures of normals.