## Gibbs Sampling Methods for Stick-Breaking Priors

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@MISC{Ishwaran_gibbssampling,

author = {Hemant Ishwaran and Lancelot F. James},

title = {Gibbs Sampling Methods for Stick-Breaking Priors},

year = {}

}

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### Abstract

... In this paper we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies to stick-breaking priors with a known P'olya urn characterization; that is priors with an explicit and simple prediction rule. Our second method, the blocked Gibbs sampler, is based on a entirely different approach that works by directly sampling values from the posterior of the random measure. The blocked Gibbs sampler can be viewed as a more general approach as it works without requiring an explicit prediction rule. We find that the blocked Gibbs avoids some of the limitations seen with the Polya urn approach and should be simpler for non-experts to use.

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Citation Context ...the best of our knowledge) and it provides a uni� ed way to connect a collection of seemingly unrelated measures scattered throughout the literature. These include (a) the Ferguson Dirichlet process=-= (Ferguson 1973, 19-=-74), (b) the two-parameter Poisson– Dirichlet process (Pitman and Yor 1997), (c) Dirichletmultinomial processes (Muliere and Secchi 1995), m-spike models (Liu 1996), � nite dimensional Dirichlet p... |

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Citation Context ...llection of seemingly unrelated measures scattered throughout the literature. These include (a) the Ferguson Dirichlet process (Ferguson 1973, 1974), (b) the two-parameter Poisson– Dirichlet process=-= (Pitman and Yor 1997),-=- (c) Dirichletmultinomial processes (Muliere and Secchi 1995), m-spike models (Liu 1996), � nite dimensional Dirichlet priors (Ishwaran and Zarepour 2000b,c) and (d) beta two-parameter processes (Is... |

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Citation Context ...e amount of research interest. (see Pitman 1995, 1996a,b, 1997, 1999; Kerov 1995; Mekjian and Chase 1997;sIshwaran and James: Gibbs Sampling for Stick-Breaking Priors 163 Zabell 1997; Tsilevich 1997; =-=Carlton 1999-=-). However, because most of this literature has appeared outside of statistics, it has gone largely unnoticed that the process possesses several properties that make it potentially useful as a Bayesia... |

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Citation Context ...of a considerable amount of research interest. (see Pitman 1995, 1996a,b, 1997, 1999; Kerov 1995; Mekjian and Chase 1997;sIshwaran and James: Gibbs Sampling for Stick-Breaking Priors 163 Zabell 1997; =-=Tsilevich 1997-=-; Carlton 1999). However, because most of this literature has appeared outside of statistics, it has gone largely unnoticed that the process possesses several properties that make it potentially usefu... |

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2 |
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Citation Context ...ween the Dirichlet process and beta random variable stick-breaking procedures. However, it was not until Sethuraman (1994) that these connections were formalized (also see Sethuraman and Tiwari 1982, =-=Donnelly and Joyce 1989,-=- and Perman, Pitman, and Yor 1992). Let â k = E 1 + + E k, where E k are iid exp415 random variables. Sethuraman (1994) established the following remarkable identity showing that the Dirichlet proces... |

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