Gibbs Sampling Methods for Stick-Breaking priors

Hemant Ishwaran and Lancelot F. James2001

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

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.

Introduction (cont’d)

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.

0 1v1

1-v1

(1-v1)(1-v2)

v2(1-v1) v3(1-v1) (1-v2)…

),Beta(~ kkk baV

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:

1

),(n

Zn nQPy

abanabaWn ),1,0[),,1Beta(~

kabbaa kk ,1

Generalized Dirichlet Random Weights

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

1

1

12111

1

,1,...2,),Beta(~,)1()1)(1(,N

kkN

kkkkkk

pp

NkbaVVVVVpVp

f(p1,..,pN)=f(pN | pN-1,…, p1) f(pN-1 | pN-2,…, p1)…f(p1)

ak=k, bk=k+1+…+N

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.

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

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 InferenceIteratively 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.