Posterior sampling from \(\epsilon\)-approximation of normalized completely random measure mixtures.

*(English)*Zbl 1358.62034Summary: This paper adopts a Bayesian nonparametric mixture model where the mixing distribution belongs to the wide class of normalized homogeneous completely random measures. We propose a truncation method for the mixing distribution by discarding the weights of the unnormalized measure smaller than a threshold. We prove convergence in law of our approximation, provide some theoretical properties, and characterize its posterior distribution so that a blocked Gibbs sampler is devised. The versatility of the approximation is illustrated by two different applications. In the first the normalized Bessel random measure, encompassing the Dirichlet process, is introduced; goodness of fit indexes show its good performances as mixing measure for density estimation. The second describes how to incorporate covariates in the support of the normalized measure, leading to a linear dependent model for regression and clustering.

##### MSC:

62G05 | Nonparametric estimation |

62F15 | Bayesian inference |

60G57 | Random measures |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |