Description
1. (10 pts.) In general, the probability that it rains on Saturday is 25%.
Weekend rain has the following relationships:
• If it rains on Saturday, the probability that it rains on Sunday is 50%.
• If it does not rain on Saturday, the probability that it rains on Sunday is 25%.
Given that it rained on Sunday, what is the probability that it rained on Saturday?
2. (20 pts.) A bug stands on a random point of the lattice below. Each point is equally likely to be the
starting point.
Every minute, the bug selects an adjacent point at random and moves to it. Each adjacent point is
equally likely to be chosen. For example, if the bug is on point B, then each probability to move to the
points A, C, or G is 1
3
.
What is the probability that the bug reaches point A in 2 moves or less? Each point is equally likely to
be the bug’s starting point. Also, assume starting at A will ”reach” the point in 0 moves.
3. (40 pts.) The idea for the maximum likelihood estimate (MLE) is to find the value of the parameter(s) for which the data has the highest probability. You are going to do this with the densities.
Suppose the 1-dimension data points x1, x2, …xn given in ”data.txt” file are drawn from a normal(gauss)
N(µ, σ2
) distribution, where µ and σ are unknown.
• (20 pts.) Formulate the likelihood function and derive the equation to find the maximum likelihood
estimate for the pair (µ, σ2
).
• (20 pts.) Implement (write the code) MLE in Matlab or Python language and provide your
plot that is similar to Figure 1. You are not allowed use any built-in functions except histogram
functions to provide you a quick of the distribution of the data.
Page 1 of 3
BLG 454E Learning From Data Homework #1
0 2 4 6 8 10 12 14 16 18
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16 MLE results: mu = 10.06, std = 2.57
MLE fixed distribution
data
Figure 1: Data and fixed gaussian distribution with MLE
4. (30 pts.) In the Table 1 below, x1, x2, x3 and xi ∈ {0, 1}, i = 1, 2, 3. xi represent the i feature vector
and y ∈ {+, −} represents the class label.
(a) (15 pts.) Construct the Naive Bayes classifier for the given training dataset in Table 1.
Hint: Estimate the class conditional prob. for each feature vector x1, x2, x3
(b) (5 pts.) Predict the class label for (x1 = 1, x2 = 1, x3 = 1) data using trained Naive Bayes approach
in part (a)
(c) (10 pts.) Calculate the probabilities of P(x1 = 1), P(x2 = 1), and P(x1 = 1, x2 = 1). Decide
whether x1 and x2 are independent or not.
Table 1: Training set for question 4
Instance x1 x2 x3 y
1 0 0 1 –
2 1 0 1 +
3 0 1 0 –
4 1 0 0 –
5 1 0 1 +
6 0 0 1 +
7 1 1 0 –
8 0 0 0 –
9 0 1 0 +
10 1 1 1 +
Submission Policy
• Prepare the report and code. Only electronic submissions through Ninova will be accepted no later
than March, 07 at 10pm.
• You may discuss the problems at an abstract level with your classmates, but you should not share
or copy code from your classmates or from the Internet. You should submit your own, individual
homework.
• Academic dishonesty, including cheating, plagiarism, and direct copying, is unacceptable.
• Note that your codes and reports will be checked with the plagiarism tools!
Page 2 of 3
BLG 454E Learning From Data Homework #1
• If a question is not clear, please let the teaching assistants know by email (kivrakh@itu.edu.tr
or cebeci16@itu.edu.tr).
Bonus marks (10pts)
• Clarity and nicely described report
• Using Latex template for the report that is given to you
Deductions (-10pts)
• Spelling errors.
• Messiness
• Lack of content.
• Irrelevant / mistaken content.
Page 3 of 3 End of homework.
