## Gibbs Sampling Methods for Stick-Breaking Priors

### Cached

### Download Links

Citations: | 225 - 17 self |

### BibTeX

@MISC{Ishwaran_gibbssampling,

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

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

year = {}

}

### Years of Citing Articles

### OpenURL

### 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.

### Citations

750 |
A bayesian analysis of some nonparametric problems
- Ferguson
- 1973
(Show Context)
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... |

434 | Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems - Antoniak - 1974 |

424 | Bayesian Density Estimation and Inference Using Mixtures - Escobar, West - 1995 |

391 | Markov chain sampling methods for Dirichlet process mixture models - Neal - 2000 |

333 | A constructive definition of dirichlet priors - Sethuraman |

283 |
Ferguson distributions via Pólya urn schemes
- Blackwell, MacQueen
- 1973
(Show Context)
Citation Context ...observation Y n+ 1 given previous sampled values Y 1 1 : : : 1 Y n from the prior (as illustration, the prediction rule for the Dirichlet process leads to the famous Blackwell–MacQueen Pólya urn; s=-=ee Blackwell and MacQueen 1973 -=-for more discussion). Our second method, the blocked Gibbs sampler, works in greater generality in that it can be applied when the Pólya urn characterization is unknown. Thus, this method can be used... |

224 | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Annals of Probability
- Pitman, Yor
- 1997
(Show Context)
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... |

147 | Estimating mixture of dirichlet process models - MacEachern, Mueller - 1998 |

110 |
Exchangeable and partially exchangeable random partitions. Probability Theory and Related
- Pitman
- 1995
(Show Context)
Citation Context ...rs. 2. THE PITMAN–YOR PROCESS, °(a1b) The recently developed two-parameter Poisson–Dirichlet process of Pitman and Yor (1997) has been the subject 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 li... |

108 | On a class of Bayesian nonparametric estimates I: density estimates - Lo - 1984 |

106 | Estimating normal means with a conjugate style dirichlet process prior - MacEachern - 1994 |

101 | A semiparametric Bayesian model for randomised block design - Bush, MacEachern - 1996 |

93 | Hierarchical priors and mixture models, with applications in regression and density estimation - West, Müller, et al. - 1994 |

91 | Random discrete distributions - Kingman - 1975 |

78 |
Nonparametric Bayes estimators based on beta processes in models for life history data
- Hjort
- 1990
(Show Context)
Citation Context ...ichlet process in censored data problems, and by Ishwaran and Zarepour (2000a), who updated approximate Dirichlet processes in hierarchical models. Also see the literature regarding the beta process (=-=Hjort 1990-=-) in survival analysis for further examples of Bayesian computational methods that include the prior in the update (for example, see Damien, Laud, and Smith 1996 and Laud, Damien, and Smith 1998). 1.1... |

75 | On the asymptotic behavior of Bayes estimates in the discrete - Freedman - 1963 |

70 | Modelling heterogeneity with and without the dirichlet process. Scandinavian journal of statistics - Green, Richardson - 2001 |

63 | Yor,Size-biased sampling of Poisson point processes and excursions - Perman, Pitman, et al. - 1992 |

52 |
Nonparametric hierarchical Bayes via sequential imputations
- Liu
- 1996
(Show Context)
Citation Context ...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 (Ishwaran and Zarepour 2000a). (This list is by no means complete, and we anticipate that mo... |

41 | On a Class of Bayesian Nonparametric Estimates - Lo - 1984 |

36 | Bayesian nonparametric estimation for incomplete data via successive substitution sampling - DOSS - 1994 |

35 | Concepts of independence for proportions with a generalization of the Dirichlet distribution - Connor, Mosimann - 1969 |

34 | Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models - Ishwaran, Zarepour - 2000 |

29 | A representation of independent increment processes without Gaussian components - Ferguson, Klass - 1972 |

28 | 2002: Approximate dirichlet process computing in finite normal mixtures: Smoothing and prior information - Ishwaran, James |

28 | Approximating distributions of random functionals of Ferguson-Dirichlet priors - Muliere, Tardella - 1998 |

28 |
Convergence of Dirichlet Measures and the Interpretation of their Parameter
- Sethuraman, Tiwari
- 1982
(Show Context)
Citation Context ...rk that drew comparisons between 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 ... |

22 | Estimating the means of several normal populations by nonparametric estimation of the distribution of the means - Escobar - 1988 |

21 | A model for the distribution of individuals by species in an environment - McCloskey - 1965 |

19 | Continuity and weak convergence of ranked and sizebiased permutations on the infinite simplex. Stochastic Process - Donnelly, Joyce - 1989 |

14 | Asymptotic behavior of Bayes estimates - Fabius - 1964 |

13 |
Coherent random allocations, and the Ewens-Pitman formula
- Kerov
- 2005
(Show Context)
Citation Context ...a1b) The recently developed two-parameter Poisson–Dirichlet process of Pitman and Yor (1997) has been the subject 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 ... |

12 | Implementation of Bayesian nonparametric inference based on beta processes - DAMIEN, LAUD, et al. - 1996 |

9 |
Applications of the two-parameter Poisson-Dirichlet distribution
- Carlton
- 1999
(Show Context)
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... |

9 | A note on a proper Bayesian bootstrap - Muliere, Secchi - 1995 |

8 | Bayesian nonparametric and covariate analysis of failure time data, in: Practical Nonparametric and Semiparametric Bayesian Statistics - Laud, Damien, et al. - 1998 |

7 | Markov Chain Monte Carlo - Ishwaran, Zarepour - 2000 |

6 | Sampling methods for bayesian nonparametric inference involving stochastic processes - Walker, Damien - 1998 |

5 | Random alms - Halmos - 1944 |

4 | A constructive de nition of dirichlet priors - Sethuraman - 1991 |

4 |
Distribution of the Mean Value for Certain Random Measures
- Tsilevich
- 1997
(Show Context)
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... |

4 | Population models with a non-parametric random coefficient distribution - Wakefield, Walker - 1994 |

4 | Disordered systems, power laws and random processes - Mekjian, Chase - 1997 |

3 | Size-Biased Sampling - Perman, Pitman, et al. - 1992 |

2 |
Continuity and Weak Convergence of Ranked and
- Donnelly, Joyce
- 1989
(Show Context)
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... |

2 | sampling stick-breaking priors - Ishwaran, Gibbs - 1973 |

1 | Concepts of Independenc e for Proportions With a Generalization of the Dirichlet Distribution - Connor, Mosimann - 1969 |

1 | A Representation of - Ferguson, Klass - 1972 |

1 | Sampling for Stick-Breaking Priors 173 Muliere - Ishwaran, Gibbs - 1995 |

1 | sampling stick-breaking priors 21 - Ishwaran, Gibbs - 2000 |