Nonparametric Bayesian Methods

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Nonparametric Bayesian Methods

• Borrows from
• Tommi Jaakolas
• http//www.ai.mit.edu/courses/6.867-f01/lectures/l
  ecture-14.pdf
• madigan_at_stat.rutgers.edu

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ibrahim 21
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Bayesian Nonparametrics

• A Dirichlet Process (DP) is a distribution on
  distributions
• Suppose F is a distribution distributed according
  to a DP, p DP(a, p0), where p0 is a probability
  distribution and a gt0
• DP has the property that (p(B1),, p(Bk)) is
  distributed according to a regular Dirichlet
  distribution
• D(a p0(B1),, a p0(Bk)) for any partition
  (B1,,Bk) of the sample space
• p0(Bj) denotes the probability contained in Bj
  under the distribution p0

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Bayesian Updating

• Suppose (x1,,xn) is a sample from F() and
  suppose F is distributed a priori F DP(a, F0),
  then
• where Fn is the empirical cdf

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DP hyperpriors

• Consider the following setup
• Then marginal distribution of x is a mixture of
  normal distributions with a random number of
  components between 1 and n

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DP hyperpriors
probability that a new value of (q,s2) occurs

• Gibbs sampling here gives different number of
  components at each simulation step

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Distribution of a Future x?