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Unformatted text preview: Introduction to Gibbs Sampling October 8, 2010 Readings: Hoff 6 October 7, 2010 Monte Carlo Sampling I We have seen that Monte Carlo sampling is a useful tool for sampling from prior and posterior distributions I By limiting attention to conjugate prior distributions, all models have had tractable posterior distributions so sampling was not really necessary (although convenient) I What if we want to use a nonconjugate prior distribution? I What if we cannot sample from the joint posterior distribution? SemiConjugate Examples Normal sampling model Y i  , iid N( , 1 ) But now assume that is independent of a priori: N( m , 1 / p ) G ( / 2 , s 2 / 2) Posterior Distribution: p ( ,  Y ) Y j p ( y i  , ) p ( ) p ( ) Factorization of Joint p ( ,  Y ) n / 2 exp 2 s 2 ( n 1) / 2 1 exp( s 2 ) exp 2 n ( y ) 2 exp( p (  m ) 2 ) = p (  , Y ) p (  Y ) = p (  Y ) p (  , Y ) First Factorization For  , Y complete the square to show that  , Y N n y + p m n + p , ( n + p ) 1 Can we recognize the marginal distribution for ? No! Second Factorization p ( ,  Y ) n / 2 exp 2 s 2 ( n 1) / 2 1 exp( s 2 ) exp 2 n ( y ) 2 exp( p (  m ) 2 ) Can recognize  , Y G n + 2 , ( n 1) s 2 + n ( y ) 2 + s 2 2 Sampling? I Suppose we were given a value of that comes from the marginal posterior distribution (say (1) = y ) I We could then draw a value of from the conditional Gamma distribution given = (1) which would give us a draw from the joint distribution ( (1) , (1) ) I (1) could be viewed as a draw from the marginal distribution (based on the first factorization), so if we now use the conditional distribution of  , Y to draw a new (2) we have another sample from the joint distribution....
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 Fall '09

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