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Problem6_6 - STA 6934 Alla Revenko Problem 6.6 Consider a...

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STA 6934 Alla Revenko Problem 6.6 Consider a version of Independent MH based on a “bound” M on f / g that is not a uniform bound; i.e. f / g > M for some x . a) Assume an Accept-Reject algorithm uses the density g with acceptance probability f(y) / M g(y) , show that the resulting variables are generated from { } f (x) min f (x),Mg(x) , % instead of f. Indeed, f (X) f (X) f (X) P(Y y) P X y | U P X y, 1| U Mg(x) Mg(x) Mg(x) f(X) f(X) P X y, 1| U Mg(x) Mg(x) = < = < + + < < = f(X) f (X) P X y,f(X) Mg(x),U P X y,f(X) Mg(x),U Mg(x) Mg(x) f(X) P U Mg(x) < + < < = < { } { } y f (x) Mg(x) f (x) Mg(x) 0 f (x) Mg(x) y f (x) Mg(x) 0 f (x) Mg(x) 0 g(x)dx du g(x)dx du g(x)dx du g(x)dx -∞ +∞ -∞ < -∞ +∞ -∞ Ι = + Ι + = ∫ ∫ ∫ ∫ { } { } y f (x) Mg(x) y f (x) Mg(x) g(x)dx f(x) g(x)dx Mg(x) f (x) g(x)dx Mg(x) f (x) g(x)dx Mg(x) -∞ +∞ -∞ < -∞ +∞ -∞ Ι = + Ι + =
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{ } { } y f (x) Mg(x) y f (x) Mg(x) M g(x)dx f(x)dx f (x)dx f (x)dx -∞ +∞ -∞ < -∞ +∞ -∞ Ι = + Ι + =
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