## Dirichlet process and related priors

My contribution this year to MCB seminar at CREST is about nonparametric Bayes (today at 2 pm, room 14). I shall start with 1) a few words of history, then introduce 2) the Dirichlet Process by several of its numerous defining properties. I will next introduce an extension of the Dirichlet Process, namely 3) the DP mixtures, useful for applications like 4) density estimation. Last, I will show 5) posterior MCMC simulations for a density model and give some 6) reference textbooks.

**1) History**

Ferguson, T.S. (1973), A Bayesian Analysis of Some Nonparametric Problems. The Annals of Statistics, 1, 209-230.

Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. The Annals of Probability, 2 183-201.

Blackwell, D. (1973). Discreteness of Ferguson selections. The Annals of Statistics, 1 356-358.

Blackwell, D. and MacQueen, J. (1973). Ferguson distributions via Polya urn schemes. The Annals of Statistics, 1 353-355.

**2) Dirichlet Process defining properties**

Mainly based on Peter Müller’s slides of class 2 at Santa Cruz this summer.

Dirichlet distribution on finite partitions

Stick-breaking/ Sethuraman representation

Polya urn analogy for the predictive probability function

Normalization of a Gamma Process

**3) Dirichlet Process Mixtures (DPM)**

Convolution of a DP with a continuous kernel to circumvent its discretness.

**4) Density estimation with DPMs**

**5) Posterior MCMC simulation**

Based on Peter Müller’s slides of class 6.

**6) Textbooks**

Bayesian nonparametrics, 2003, J. K. Ghosh, R. V. Ramamoorthi.

Bayesian nonparametrics, 2010, Nils Lid Hjort, Chris Holmes, Peter Müller, Stephen G. Walker, and contributions by Subhashis Ghosal, Antonio Lijoi, Igor Prünster, Yee Whye Teh, Michael I. Jordan, Jim Griffin, David B. Dunson, Fernando Quintana.

Paulsaid, on 5 October 2010 at 05:23The first chapter of Schervish’s “Theory of Statistics” contains excelent materials on the Dirichlet Process.

Julyansaid, on 8 October 2010 at 11:32Thank you Paul for this reference. The Lemma on the marginal distribution of observations from a DP is particularly useful!

Dirichlet Process for dummies « Statisfactionsaid, on 16 November 2010 at 11:24[…] mentionned in an earlier post, a foundational paper and still a nice reading today, which introduced the DP, is Ferguson, T.S. […]

Priors on probability measures « Statisfactionsaid, on 24 April 2012 at 20:36[…] work on Projective limit random probabilities on Polish spaces. It will follow my previous presentation about Bayesian nonparametrics on the Dirichlet […]