Mixtures of generalized gamma convolution processes

A class of random measures derived from generalized gamma convolutions (GGC) is developed in the context of Bayesian statistics applications. These random measures can be represented as functionals of gamma processes, specifically as marginalized weighted gamma processes, offering greater flexibility compared to traditional gamma processes. Furthermore, they can be expressed in terms of a Dirichlet process, utilizing the approach used to represent weighted gamma processes as Dirichlet processes. Over the years, a variety of sampling strategies for Dirichlet process mixture models have been introduced, making it straightforward to sample posterior quantities derived from GGC processes. In this work, we derive Blackwell and MacQueen’s (1973) Pólya urn formula and the Chinese restaurant process from GGC processes. These results have direct applications in Bayesian mixture model estimation, and numerical simulations are provided for illustration.