Generalized weighted Chinese restaurant processes for species sampling mixture models.

*(English)*Zbl 1086.62036Summary: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently, the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with J. Pitman [Probab. Theory Relat. Fields 102, No. 2, 145–158 (1995; Zbl 0821.60047)], we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of A. Y. Lo [Ann. Stat. 12, 351–357 (1984; Zbl 0557.62036)] for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work of L. J. Brunner et al. [Weighted Chinese restaurant processes and Bayesian mixture models. Unpublished manuscript (2001)] by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models.

We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors.

We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors.

##### MSC:

62F15 | Bayesian inference |

62G99 | Nonparametric inference |

60G09 | Exchangeability for stochastic processes |

65C60 | Computational problems in statistics (MSC2010) |

62L99 | Sequential statistical methods |