Gibbs Sampling Methods for Stick-Breaking priors

1. 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

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

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

4. Introduction (cont’d) v2(1-v1) v3(1-v1) (1-v2) … 0 1 v1 1-v1 (1-v1)(1-v2) • Steak-breaking construction: • , i.i.d. random variables. • N is finite: set VN=1 to guarantee . • pk 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.

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

6. Generalized Dirichlet Random Weights ak=k, bk=k+1+…+N • Finite stick-breaking priors & GD: • Random weights p=[p1,..,pN] constructed from a finite Stick-breaking procedure is a Generalized Dirichlet distribution (GD). The density for p is f(p1,..,pN)=f(pN | pN-1,…, p1) f(pN-1 | pN-2,…, p1)…f(p1)

7. Generalized Dirichlet Random Weights • Finite dimensional Dirichlet priors: • A random measure with weights, p=(p1,…,pN)~Dirichlet(1,…, N), • p has a GD distribution w/ ak=k, bk=k+1+…+N. • Connection: all random measures based on Dirichlet random weights are Stick-breaking random measure w/ finite N.

8. Truncations • Finite Stick-breaking random measure can be a truncation of . • Discard the N+1, N+2,… terms in , and replace pN with 1-p1-…-pN-1. • It’s an approximation. • When as a prior is applied in Bayeisan hierarchical model, the Bayesian marginal density under the truncation is

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

10. 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)

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

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

13. 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 ndata 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.

14. Results Simulated 50 observations from a standard normal distribution.