Gibbs Sampling Methods for Stick-Breaking priors Hemant Ishwaran and Lancelot F. James 2001 Presented by Yuting Qi ECE Dept., Duke Univ. 03/03/06

Overview Introduction What’s Stick-breaking priors? Relationship between different priors Two Gibbs samplers Polya Urn Gibbs sampler Blocked Gibbs sampler Results Conclusions

Introduction What’s Stick-Breaking Priors? Discrete random probability measures p k : random weights, independent of Z k, Z k are iid random elements with a distribution H, where H is nonatomic.. Random weights are constructed through stick-breaking procedure.

Introduction (cont’d) Steak-breaking construction:, i.i.d. random variables. N is finite: set V N =1 to guarantee. p k have the generalized Dirichlet distribution which is conjugate to multinomial distribution. N is infinite: Infinite dimensional priors include the DP, two-parameter Poisson-Dirichlet process (Pitman-Yor process), and beta two-parameter process. 01 v1v1 1-v 1 (1-v 1 )(1-v 2 ) v 2 (1-v 1 )v 3 (1-v 1 ) (1-v 2 ) …

Pitman-Yor Process, Two-parameter Poisson-Dirichlet Process: Discrete random probability measures Q n have a GEM distribution Prediction rule (Generalized Polya Urn characterization): A special case of Stick-breaking random measure:

Generalized Dirichlet Random Weights Finite stick-breaking priors & GD: Random weights p=[p 1,..,p N ] constructed from a finite Stick-breaking procedure is a Generalized Dirichlet distribution (GD). The density for p is f(p 1,..,p N )=f(p N | p N-1,…, p 1 ) f(p N-1 | p N-2,…, p 1 )…f(p 1 ) a k =  k, b k =  k+1 +…+  N

Generalized Dirichlet Random Weights Finite dimensional Dirichlet priors: A random measure with weights, p=(p 1,…,p N )~Dirichlet(  1,…,  N ), p has a GD distribution w/ a k =  k, b k =  k+1 +…+  N. Connection: all random measures based on Dirichlet random weights are Stick-breaking random measure w/ finite N.

Truncations Finite Stick-breaking random measure can be a truncation of. Discard the N+1, N+2,… terms in, and replace p N with 1-p 1 -…-p N-1. It’s an approximation. When as a prior is applied in Bayeisan hierarchical model, the Bayesian marginal density under the truncation is

Truncations (cont’d) If n=1000, N=20,  =1, then ~10^(-5)

Polya Urn Gibbs Sampler Stick-breaking measures used as priors in Bayesian semiparametric models, Integrating over P, we have Polya Urn Gibbs sampler: (a) (b)

Blocked Gibbs Sampler Assume the prior is a finite dimensional, the model is rewritten as Direct Posterior Inference Iteratively draw values Values from joint distribution of Each draw defines a random measure

Blocked Gibbs Algorithm Algorithm: Let denote the set of current m unique values of K,

Comparisons In Polya Urn Process, in one Gibbs iteration, each data inquires existing m clusters & a new cluster one by one. The extreme case is each data belongs to one cluster, ie, # of cluster equals to # of data points. In Blocked Gibbs sampler, in one Gibbs iteration, all n data points inquire existing m clusters & N-m new different clusters. That’s the infinite un-present clusters in Polya Urn process is represented by N-m clusters in Blocked Gibbs sampler. Since # of data points is finite, once N >= n, N possible clusters are enough for all data even in the extreme case where each data belongs to one cluster. In this sense, Blocked Gibbs sampler is equivalent to Polya Urn Gibbs sampler.

Results Simulated 50 observations from a standard normal distribution.