Posterior sampling from truncated Ferguson-Klass representation of normalised completely random measure mixtures

In this paper, we study the finite approximation of the completely random measure (CRM) by truncating its Ferguson-Klass representation. The approximation is obtained by keeping the N largest atom weights of the CRM unchanged and combining the smaller atom weights into a single term. We develop the simulation algorithms for the approximation and characterise its posterior distribution, for which a blocked Gibbs sampler is devised. We demonstrate the usage of the approximation in two models. The first assumes such an approximation as the mixing distribution of a Bayesian nonparametric mixture model and leads to a finite approximation to the model posterior. The second concerns the finite approximation to the Caron-Fox model. Examples and numerical implementations are given based on the gamma, stable and generalised gamma processes.