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.

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.