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MARKOVCHAIN USING MONTECARLO

# MARKOVCHAIN USING MONTECARLO - Markov Chain Monte Carlo...

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Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Luke Tierney Department of Statistics & Actuarial Science University of Iowa

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Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Basic Ratio of Uniforms Method Introduced by Kinderman and Monahan (1977) . Suppose f is a (possibly unnormalized) density Suppose V , U are uniform on A = { ( v , u ) : 0 < u < f ( v / u ) } Then X = V / U has density (proportional to) f ( x ) 1
Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Example: Standard Normal Distribution Suppose f ( x ) = 1 2 π e - x 2 / 2 Then A is bounded. Can use rejection sampling from an enclosing rectangle. -0.5 0.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 v u 2

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Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Some Generalizations If f ( x ) = f 0 ( x ) f 1 ( x ) and V , U has density proportional to f 0 ( v / u ) on A = { ( v , u ) : 0 < u < f 1 ( v / u ) } then X = V / U has density proportional to f If V , U are uniform on A = { ( v , u ) : 0 < u < f ( v / u + μ ) } then X = V / U + μ has density proportional to f 3
Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Some Generalizations (cont.) Choosing μ as the mode of f often works well. For a Gamma( α = 50 ) density f ( x ) x 49 e - x : 0 2 4 6 8 10 12 0.00 0.05 0.10 0.15 0.20 v u -1.0 -0.5 0.0 0.5 1.0 1.5 0.00 0.05 0.10 0.15 0.20 v u μ = 0 μ = α - 1 = 49 4

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Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Higher Dimensions Stef˘anescu and V˘aduva (1987); Wakefield, Gelfand and Smith (1991) If V , U are uniform on A = { ( v , u ) : v R d , 0 < u < d + 1 f ( v / u + μ ) } then X = V / U + μ has density proportional to f Can rejection sample from hyper-rectangle Usually not practical in higher dimensions Alternative: Sample A by MCMC 5
Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Higher Dimensions (cont.) Some Regions for d = 2 : Bivariate Normal, ρ = 0 . 73 Variance Components 6

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Markov Chain Monte Carlo Using the Ratio-of-Uniforms Transformation Bormio 2005 Some Properties of the Region A A is bounded if f ( x ) and x d + 1 f ( x ) are bounded.
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MARKOVCHAIN USING MONTECARLO - Markov Chain Monte Carlo...

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