Consider (X1, X2, X3) with the joint pdf f(x1, 12, 13) x exp{-} (x² + x² + x} - {₁x2-x2x3)}. The full conditionals for X₂X₁ = 1, X3 = 23 and X3|X₁ = ₁, X2 = 1₂ for this joint density are X2|X₁ = x1, X3 = 23 ~ 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. (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, marginally, X₁~ N(0,3) and, conditionally, X₂X₁ = ₁ ~ N (₁,3) while, as above, X3|X₁ = 21, X₂ = x2 ~ 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).
Consider (X1, X2, X3) with the joint pdf f(x1, 12, 13) x exp{-} (x² + x² + x} - {₁x2-x2x3)}. The full conditionals for X₂X₁ = 1, X3 = 23 and X3|X₁ = ₁, X2 = 1₂ for this joint density are X2|X₁ = x1, X3 = 23 ~ 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. (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, marginally, X₁~ N(0,3) and, conditionally, X₂X₁ = ₁ ~ N (₁,3) while, as above, X3|X₁ = 21, X₂ = x2 ~ 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).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
Do part b & c
Transcribed Image Text:Consider (X1, X2, X3)" with the joint pdf
f(nn,za, zs) x exp {-을 (국 + 금 + -3피2-2213)}.
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 X1X2= 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 = 1, X2 = r2~ N (¿r2, 1) .
%3D
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).
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