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531f10GIBBS1 - STAT 531 Gibbs Sampler HM Kim Department of...

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STAT 531: Gibbs Sampler HM Kim Department of Mathematics and Statistics University of Calgary Fall 2010 1/23
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The Gibbs sampler was introduced in the context of image processing by Geman and Geman, 1984 It is a special case of MH samplings The task remains to specify how to construct a Markov chain whose values converge to the target distribution. Can we know the joint distribution simply by knowing the full conditional distributions? Fall 2010 2/23
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Hammersley-Clifford Theorem (for two blocks) Suppose we have a joint density f ( x , y ). The theorem proves that we can write out the joint density in terms of the conditional densities f ( x | y ) and f ( y | x ): f ( x , y ) = f ( y | x ) R f ( y | x ) f ( x | y ) dy We can write the denominator as Z f ( y | x ) f ( x | y ) dy = Z f ( x , y ) f ( x ) f ( x , y ) f ( y ) dy = Z f ( y ) f ( x ) dy = 1 f ( x ) Fall 2010 3/23
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Then f ( x , y ) = f ( y | x ) 1 f ( x ) = f ( x | y ) f ( x ) = f ( x , y ) The theorem shows that knowledge of the conditional densities allows us to get the joint density. This works for more than two blocks of parameters. Fall 2010 4/23
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The key to the Gibbs sampler is that one only considers univariate conditional distributions . Such conditional distributions are far easier to simulate than complex joint distributions and usually have simple forms. Thus one simulates q random variables sequentially from the q univariate conditionals rather than generating a single q -dimensional vector in a single pass using the full joint distributions. Fall 2010 5/23
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, Example : To introduce the Gibbs sampler, consider a bivariate random variable ( x , y ), and suppose we wish to compute one or both marginals, p ( x ) and p ( y ).
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