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# lec26 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 14.384 Time Series Analysis Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Gibbs Sampling 1 14.384 Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe December 11, 2007 Lecture 26 MCMC: Gibbs Sampling Last time, we introduced MCMC as a way of computing posterior moments and probabilities. The idea was to draw a sample from the posterior distribution and use moments from this sample. We drew these samples by constructing a Markov Chain with the posterior distribution as its invariant measure. In particular, we found a transition kernel, P ( x, dy ), such that π ( y ) = P ( x, dy ) π ( x ) dx . The Gibbs sampler is another form of MCMC. R Gibbs Sampling Suppose we can write our random variable of interest as components, x = ( x 1 , x 2 , ..., x d ), such that we can simulate the distribution of each component conditional on the others, i.e. we can draw from π ( x k | x 1 , ..., x k- 1 , x k +1 , . ∀ k . We want to sample from the joint distribution, π ( x ). Gibbs sampling constructs a Markov Chain, x (1) → x (2) → ... , with step x ( j ) → x ( j +1) given by: Simulate ( j +1) ( j +1) ( j ) ( j ) • x 1 from π ( x 1 | x 2 , ..., x d ) ( j +1) ( j +1) ( j +1) ( j ) ( j ) • x 2 from π ( x 2 | x 1 , x 3 , ..., x d ) ( j +1) ( j +1) ( j +1) ( j +1) ( j ) ( j ) • x 3 from π ( x 3 | x 1 , x 2 , x 4 , ..., x d ) • ... Claim 1. π ( x ) is the invariant measure for this Markov Chain. Proof. The transition kernel is: P ( x, dy ) = π ( y 1 | x 2 , ..., x d ) π ( y 2 | y 1 , x 3 , ..., x d ) ...π ( y d | y 1 , ..., y d- 1 ) dy 1 ...dy d = Y π ( y k | y 1 , ..., y k- 1 , x k +1 , ..., x d ) dy k k We want to show that R P ( x, dy ) π ( x ) dx = π ( y ) dy...
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lec26 - MIT OpenCourseWare http/ocw.mit.edu 14.384 Time...

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