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Consider (X1, X2, X3) with the joint pdf f(x1, x2, 3) x exp{-} (x² + x² + x² − 3x₁2x2-x2x3)}. The full conditionals for X₂X₁ = 21, X3 = 23 and X3|X₁ = 1, X₂ = 2₂ for this joint density are X2|X1 = 21, X3 = x3 ~ N (1 + x3,7), X3|X1 = 21, X₂ = 22 ~ N(2, 1). You do not need to show how to obtain these two conditional distributions. (a) Find the full conditional for X1|X2 = 22, X3 = 23.

Consider (X1, X2, X3) with the joint pdf f(x1, x2, 3) x exp{-} (x² + x² + x² − 3x₁2x2-x2x3)}. The full conditionals for X₂X₁ = 21, X3 = 23 and X3|X₁ = 1, X₂ = 2₂ for this joint density are X2|X1 = 21, X3 = x3 ~ N (1 + x3,7), X3|X1 = 21, X₂ = 22 ~ N(2, 1). You do not need to show how to obtain these two conditional distributions. (a) Find the full conditional for X1|X2 = 22, X3 = 23.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 52RE
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Consider (X1, X2, X3)" with the joint pdf f(n,r2, Ig) x exp {-à (+ 금교 + -피-2213)}. The full conditionals for X2X1 = x1, X3 = r3 and X3|X1 =x1, X2 = r2 for this joint density are %3D %3D %3D X2|X1= 피1, X3 3Dr3 ~ N (피 + 을s, 꼭), X3|X1 = #1, X2 = r2~N (r2, 1). You do not need to show how to obtain these two conditional distributions. (a) Find the full conditional for X1|X2 = x2, X3 = r3. %3D (b) Given current values, and the ability to generate random variables from any univariate normal distribution, show how Gibbs sampling can be used to obtain the next set of sample values from the joint distribution. Make sure you explicitly give any univariate normal distributions used along the way and use the notation from the course. (c) It turns out that Gibbs sampling is not actually necessary to simulate from this joint distribution. This is because, marg X1 - N(0, ) and, conditionally, X2|X1 = #1~ N (x1,3) while, as above, X3|X1 = 21, X2 = 2 ~ N (}#2, 1) . Explain carefully how you can generate a sample of values from the joint distribution using this information (and the ability to generate random variables from any univariate normal distribution). (d) What is the major advantage of the method of part (c) over the Gibbs sampler in this case? What aspect of the current problem, which is not the case for many other multivariate distributions, is responsible for this?
Transcribed Image Text:Consider (X1, X2, X3)" with the joint pdf f(n,r2, Ig) x exp {-à (+ 금교 + -피-2213)}. The full conditionals for X2X1 = x1, X3 = r3 and X3|X1 =x1, X2 = r2 for this joint density are %3D %3D %3D X2|X1= 피1, X3 3Dr3 ~ N (피 + 을s, 꼭), X3|X1 = #1, X2 = r2~N (r2, 1). You do not need to show how to obtain these two conditional distributions. (a) Find the full conditional for X1|X2 = x2, X3 = r3. %3D (b) Given current values, and the ability to generate random variables from any univariate normal distribution, show how Gibbs sampling can be used to obtain the next set of sample values from the joint distribution. Make sure you explicitly give any univariate normal distributions used along the way and use the notation from the course. (c) It turns out that Gibbs sampling is not actually necessary to simulate from this joint distribution. This is because, marg X1 - N(0, ) and, conditionally, X2|X1 = #1~ N (x1,3) while, as above, X3|X1 = 21, X2 = 2 ~ N (}#2, 1) . Explain carefully how you can generate a sample of values from the joint distribution using this information (and the ability to generate random variables from any univariate normal distribution). (d) What is the major advantage of the method of part (c) over the Gibbs sampler in this case? What aspect of the current problem, which is not the case for many other multivariate distributions, is responsible for this?
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