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Problem7_1

# 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 | ( | ) ( | ) t X Y t t X Y t Y f x X f 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: * * | | * * | | ( , ) ( | ) ( | ) ( , ) ( | ) ( | ) Y X X Y X Y Y X K x x f y x f x y dy K y y f x y f y x dx = = ò ò So, ( X t ) and ( Y t ) also depend only on x t-1 and y t-1 respectively, so ( X t ) and ( Y t ) are Markov chains. b) * * | | * | | * * * | | ( , ) ( ) ( | ) ( | ) ( ) ( | ) ( | ) ( ) ( | ) ( , ) ( | ) ( ) ( ) X Y X X Y X X Y Y X X X Y XY X Y Y X K x x f x dx f y x f x y dy f
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