Full uncertainty analysis for Bayesian nonparametric mixture models

A full posterior analysis for nonparametric mixture models using Gibbs-type prior distributions is presented. This includes the well known Dirichlet process mixture (DPM) model. The random mixing distribution is removed enabling a simple-to-implement Markov chain Monte Carlo (MCMC) algorithm. The removal procedure takes away some of the posterior uncertainty and how it is replaced forms a novel aspect to the work. The removal, MCMC algorithm and replacement of the uncertainty only require the probabilities of a new or an old value associated with the corresponding Gibbs-type exchangeable sequence. Consequently, no explicit representations of the prior or posterior are required and instead only knowledge of the exchangeable sequence is needed. This allows the implementation of mixture models with full posterior uncertainty, not previously possible, including one introduced by Gnedin. Numerous illustrations are presented, as is an R-package called CopRe which implements the methodology, and other supplemental material.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

A full posterior analysis method for nonparametric mixture models using Gibbs-type prior distributions, including the well known Dirichlet process mixture (DPM) model, is presented. The method removes the random mixing distribution and enables a simple-to-implement Markov chain Monte Carlo (MCMC) algorithm. The removal procedure reduces some of the posterior uncertainty and introduces a novel replacement approach. The method only requires the probabilities of a new or an old value associated with the corresponding Gibbs-type exchangeable sequence, without the need for explicit representations of the prior or posterior distributions. This allows the implementation of mixture models with full posterior uncertainty, including one introduced by Gnedin. The paper also provides numerous illustrations and introduces an R-package called CopRe that implements the methodology.