1. Chinese Restaurant Representation Stick-Breaking Construction H X Yi … P  H X1 X2 XN Y1 Y2 YN
Pólya urn scheme
Inference with data
Explicit form for prior P
Integrate out P
DP prior, Pólya urn
likelihood
(X1, …, Xn) are dependent
(K1, …, Kn) are independent given p.

2. Pólya Urn Gibbs Sampler
Blocked Gibbs Sampler
Sample indicator Ki i=1,…,N,
For Ki=0, draw new Xi from
For Ki>0, generate a new set of Xk* according to posteriors
Sample indicators K from its conditional posteriors (multivariate distributions):
Sample Z:
For those j not occupied by any Xi, sample Zj from base prior H
For those j occupied by Xi, sample Zj from conditional posterior
Sample p from its conditional posteriors:
Drawbacks:
(c1,…,cN) are dependent, slowly mixing, at least N+1 component sampler, update one at a time.
Integration in qi0 may be intractable.
Inference for the posterior of P only based on the posterior Xi values.
Advantage:
Fast mixing, 3-component sampler.
Drawbacks:
Must truncate at a finite level, otherwise, the normalized constant in is an infinite sum.