Explicit form for prior P

1
Stick-Breaking Construction
Chinese Restaurant Representation

• Pólya urn scheme
• Inference with data

• Explicit form for prior P

Integrate out P

• (X1, , Xn) are dependent

• (K1, , Kn) are independent given p.

2
Blocked Gibbs Sampler
Pólya Urn Gibbs Sampler

• 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

• Sample indicator Ki i1,,N,
• For Ki0, draw new Xi from
• For Kigt0, generate a new set of Xk according to
  posteriors

• Drawbacks
• (c1,,cN) are dependent, slowly mixing, at least
  N1 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.