531f10GIBBS2

531f10GIBBS2 - 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 1/19
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Suppose we have a joint distribution p ( θ 1 , ··· , θ k ) that we want to sample from We can use the Gibbs sampler to sample from the joint distribution if we knew the full conditional distributions for each parameter. For each parameter, the full conditional distribution is the distribution of the parameter conditional on the known information and all the other parameters: p ( θ j | θ - j ) Fall 2010 2/19
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Calculating full conditional distributions 1. Write out the full target distribution ignoring constants of proportionality. 2. Pick a block of parameters (for example, θ 1 ) and drop everything that doesnt depend on θ 1 . 3. Use your knowledge of distributions to figure out what the normalizing constant is (and thus what the full conditional distribution p ( θ 1 | θ - 1 ) is). 4. Repeat steps 2 and 3 for all parameter blocks. Fall 2010 3/19
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Gibbs sampler steps Lets suppose that we are interested in sampling from the posterior p ( θ | y ), where θ is a vector of three parameters, θ 1 , θ 2 , θ 3 . The steps to a Gibbs Sampler are 1. Pick a vector of starting values θ (0) . 2. Start with any θ (order does not matter, but Ill start with θ 1 for convenience). Draw a value θ (1) 1 from the full conditional p ( θ 1 | θ (0) 2 , θ (0) 3 ) . 3. Draw a value θ (1) 2 (again order does not matter) from the full conditional p ( θ 2 | θ (1) 1 , θ (0) 3 ) . Note that we must use the updated value of θ (1) 1 . Fall 2010 4/19
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4. Draw a value θ (1) 3 from the full conditional p ( θ 3 | θ (1) 1 , θ (1) 2 ) using both updated values (step 2-4 are analogous to multiplying α (0) and P to get α (1) and then drawing θ (1) from α (1) ). 5.
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531f10GIBBS2 - STAT 531: Gibbs Sampler HM Kim Department of...

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