(Gaussian Mixture Model (GMM)) This question is about (a simplified version of) the Gaussian Mix- ture Model (GMM), which is a popular model in statistics, data science and machine learning. For example, it is used in image processing and various clustering algorithms. Suppose that K is a discrete random variable that can either be 0 or 1 with probability to and #₁ respectively, that is, K = [0, with probability To = P(K = 0), 1, with probability ₁ = P(K = 1) = 1 - To Conditional on K = k with k = {0, 1}, the distribution of X is N(μ, o), a normal distribution with mean and variance of. That is, X|K=0~ N(μo, o), X|K = 1 ~ N(μ₁, 07). (a) Derive the joint density of (X, K). State clearly the support of (X, K) in the joint density. Hint: consider conditional distribution and the law of total probability. (b) Denote the distribution of (X, K) ~ GMM(To, 0, 0, ₁,0). Note that ₁ can be omitted as a parameter since ₁=1 - 7o. Suppose that we have an i.i.d. random sample of size n of these n pairs (X₁, K₁), (X2, K₂),..., (Xn, Kn). Each X; belongs to either group 0 or group 1 depending on K₁. Using part (a), derive the maximum likelihood estimator for all the five parameters To, Ho, ₁,0. Hint: Let no = -1 1{K₁=0} and n₁ = 11{K₁=1} be the number of X; that belongs to group 0 and group 1 respectively. You may find expressing the likelihood function in terms of no and n₁ useful.
(Gaussian Mixture Model (GMM)) This question is about (a simplified version of) the Gaussian Mix- ture Model (GMM), which is a popular model in statistics, data science and machine learning. For example, it is used in image processing and various clustering algorithms. Suppose that K is a discrete random variable that can either be 0 or 1 with probability to and #₁ respectively, that is, K = [0, with probability To = P(K = 0), 1, with probability ₁ = P(K = 1) = 1 - To Conditional on K = k with k = {0, 1}, the distribution of X is N(μ, o), a normal distribution with mean and variance of. That is, X|K=0~ N(μo, o), X|K = 1 ~ N(μ₁, 07). (a) Derive the joint density of (X, K). State clearly the support of (X, K) in the joint density. Hint: consider conditional distribution and the law of total probability. (b) Denote the distribution of (X, K) ~ GMM(To, 0, 0, ₁,0). Note that ₁ can be omitted as a parameter since ₁=1 - 7o. Suppose that we have an i.i.d. random sample of size n of these n pairs (X₁, K₁), (X2, K₂),..., (Xn, Kn). Each X; belongs to either group 0 or group 1 depending on K₁. Using part (a), derive the maximum likelihood estimator for all the five parameters To, Ho, ₁,0. Hint: Let no = -1 1{K₁=0} and n₁ = 11{K₁=1} be the number of X; that belongs to group 0 and group 1 respectively. You may find expressing the likelihood function in terms of no and n₁ useful.
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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Transcribed Image Text:(Gaussian Mixture Model (GMM)) This question is about (a simplified version of) the Gaussian Mix-
ture Model (GMM), which is a popular model in statistics, data science and machine learning. For
example, it is used in image processing and various clustering algorithms. Suppose that K is a discrete
random variable that can either be 0 or 1 with probability to and ₁ respectively, that is,
K =
[0, with probability To = P(K = 0),
1, with probability #₁ = P(K = 1) = 1 - To
Conditional on K = k with k = {0, 1}, the distribution of X is N(µ,02), a normal distribution with
mean and variance of. That is,
X|K=0~ N(μo, o),
X|K = 1 ~ N(μ₁, 07).
(a) Derive the joint density of (X, K). State clearly the support of (X, K) in the joint density. Hint:
consider conditional distribution and the law of total probability.
(b) Denote the distribution of (X, K) ~ GMM (To, Ho, o, ₁,0). Note that ₁ can be omitted as a
parameter since ₁=1 o. Suppose that we have an i.i.d. random sample of size n of these n
pairs (X₁, K₁), (X₂, K₂),..., (Xn, Kn). Each X; belongs to either group 0 or group 1 depending
on K₁. Using part (a), derive the maximum likelihood estimator for all the five parameters
To Mo, ₁,0. Hint: Let no = Σi=1¹{K;=0} and n₁ = Σï=1¹{K;=1} be the number of X; that
belongs to group 0 and group 1 respectively. You may find expressing the likelihood function in
terms of no and n₁ useful.
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