Problem7_1 - STA 6934 Problem 7.1 Vladimir Boginski...

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STA 6934 Problem 7.1 Vladimir Boginski Consider the Gibbs sampler (7.1.1): Set X 0 =x 0 , for t = 1,2,…, generate | 1 | ( | ) ( | ) tX Yt Yf x Xf y : : where f X|Y and f Y|X are the conditional distributions. a) From the description above, it is easy to see that the pair ( X t ,Y t ) depends only on the pair ( x t-1 ,y t-1 ), so ( X t ,Y t ) is a Markov chain. The transition kernels for ( X t )and( Y t )are: ** || (, ) ( | )( | ) ( | | ) YX XY Kxx f y xf x ydy Kyy f x yf y xd x = = ò ò So, ( X t Y t )alsodependonlyon x t-1 and y t-1 respectively, so ( X t Y t )areMarkov chains. b) * * (, ) () ( | ) () (|) ( | ) ( | , ) ( | ) X X Y X X Y X Kxx f xdx f x ydyf xdx f x y f y x f x dxdy fx yf x y d x d y y f y d y f x == = òò ò ò
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This note was uploaded on 01/15/2012 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.

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