MAST30027: Modern Applied Statistics
Assignment 3, 2022.
Due: 11:59pm Sunday October 2rd
• This assignment is worth 14% of your total mark.
• To get full marks, show your working including 1) R commands and outputs you use, 2)
mathematics derivation, and 3) rigorous explanation why you reach conclusions or answers.
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1. The file assignment3 prob1.txt contains 300 observations. We can read the observations
and make a histogram as follows.
> X = scan(file="assignment3_prob1.txt", what=double())
Read 300 items
> length(X)
[1] 300
> hist(X)
We will model the observed data using a mixture of three binomial distributions. Specifically,
we assume the observations X1, . . . , X300 are independent to each other, and each Xi follows
this mixture model:
Zi ∼ categorical (pi1, pi2, 1− pi1 − pi2),
Xi|Zi = 1 ∼ Binomial(20, p1),
Xi|Zi = 2 ∼ Binomial(20, p2),
Xi|Zi = 3 ∼ Binomial(20, p3).
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The binomial distribution has probability mass function
f(x;m, p) =
(
m
x
)
px(1− p)m−x.
We aim to obtain MLE of parameters θ = (pi1, pi2, p1, p2, p3) using the EM algorithm.
(a) (5 marks) Let X = (X1, . . . , X300) and Z = (Z1, . . . , Z300). Derive the expectation
of the complete log-likelihood, Q(θ, θ0) = EZ|X,θ0 [log(P (X,Z|θ))].
(b) (3 marks) Derive E-step of the EM algorithm.
(c) (5 marks) Derive M-step of the EM algorithm.
(d) (5 marks) Note: Your answer for this problem should be typed. Answers
including screen-captured R codes or figures won’t be marked.
Implement the EM algorithm and obtain MLE of the parameters by applying the imple-
mented algorithm to the observed data, X1, . . . , X300. Set EM iterations to stop when either
the number of EM-iterations reaches 100 (max.iter = 100) or the incomplete log-likelihood
has changed by less than 0.00001 ( = 0.00001). Run the EM algorithm two times with
the following two different initial values and report estimators with the highest incomplete
log-likelihood.
pi1 pi2 p1 p2 p3
1st initial values 0.3 0.3 0.2 0.5 0.7
2nd initial values 0.1 0.2 0.1 0.3 0.7
For each EM run, check that the incomplete log-likelihoods increase at each EM-step by
plotting them.
2. The file assignment3 prob2.txt contains 100 observations. We can read the 300 observa-
tions from the problem 1 and the new 100 observations and make histograms as follows.
> X = scan(file="assignment3_prob1.txt", what=double())
Read 300 items
> X.more = scan(file="assignment3_prob2.txt", what=double())
Read 100 items
> length(X)
[1] 300
> length(X.more)
[1] 100
2
> par(mfrow=c(2,2))
> hist(X, xlim=c(0,20), ylim=c(0,80))
> hist(X.more, xlim=c(0,20), ylim=c(0,80))
> hist(c(X,X.more), xlim=c(0,20), ylim=c(0,80), xlab="X + X.more", main = "Histogram of X + X.more")
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LetX1, . . . , X300 andX301, . . . , X400 denote the 300 observations from assignment3 prob1.txt
and the 100 observations from assignment3 prob2.txt, respectively. We assume the ob-
servations X1, . . . , X400 are independent to each other. We model X1, . . . , X300 (from
assignment3 prob1.txt) using the mixture of three binomial distributions (as we did in the
problem 1), but we model X301, . . . , X400 (from assignment3 prob2.txt) using one of the
three binomial distributions. Specifically, for i = 1, . . . , 300, Xi follows this mixture model:
Zi ∼ categorical (pi1, pi2, 1− pi1 − pi2),
Xi|Zi = 1 ∼ Binomial(20, p1),
Xi|Zi = 2 ∼ Binomial(20, p2),
Xi|Zi = 3 ∼ Binomial(20, p3),
and for i = 301, . . . , 400,
Xi ∼ Binomial(20, p1).
We aim to obtain MLE of parameters θ = (pi1, pi2, p1, p2, p3) using the EM algorithm.
(a) (5 marks) Let X = (X1, . . . , X400) and Z = (Z1, . . . , Z300). Derive the expectation
of the complete log-likelihood, Q(θ, θ0) = EZ|X,θ0 [log(P (X,Z|θ))].
(b) (5 marks) Derive E-step and M-step of the EM algorithm.
(c) (5 marks) Note: Your answer for this problem should be typed. Answers
including screen-captured R codes or figures won’t be marked.
Implement the EM algorithm and obtain MLE of the parameters by applying the imple-
mented algorithm to the observed data, X1, . . . , X400. Set EM iterations to stop when either
the number of EM-iterations reaches 100 (max.iter = 100) or the incomplete log-likelihood
has changed by less than 0.00001 ( = 0.00001). Run the EM algorithm two times with
the following two different initial values and report estimators with the highest incomplete
log-likelihood.
pi1 pi2 p1 p2 p3
1st initial values 0.3 0.3 0.2 0.5 0.7
2nd initial values 0.1 0.2 0.1 0.3 0.7
For each EM run, check that the incomplete log-likelihoods increase at each EM-step by
plotting them.
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