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10-601 Machine Learning: Homework 3
Due 5 p.m. Wednesday, February 4, 2015
Instructions
• Late homework policy: Homework is worth full credit if submitted before the due date, half credit
during the next 48 hours, and zero credit after that. You must turn in at least n−1 of the n homeworks
to pass the class, even if for zero credit.
• Collaboration policy: Homeworks must be done individually, except where otherwise noted in the
assignments. “Individually” means each student must hand in their own answers, and each student
must write and use their own code in the programming parts of the assignment. It is acceptable for
students to collaborate in figuring out answers and to help each other solve the problems, though you
must in the end write up your own solutions individually, and you must list the names of students you
discussed this with. We will be assuming that, as participants in a graduate course, you will be taking
the responsibility to make sure you personally understand the solution to any work arising from such
collaboration.
• Online submission: You must submit your solutions online on autolab. We recommend that you
use LATEX to type your solutions to the written questions, but we will accept scanned solutions as well.
On the Homework 3 autolab page, you can download the template, which is a tar archive containing a
blank placeholder pdf for the written question and one Octave source file for each of the programming
questions. Replace the pdf file with one that contains your solutions to the written questions (g), (h),
and (i) (include all parts in the same pdf) and fill in each of the Octave source files. When you are
ready to submit, create a new tar archive of the top-level directory and submit your archived solutions
online by clicking the “Submit File” button. You should submit a single tar archive identical to the
template, except with each of the Octave source files filled in and with the blank pdf replaced by your
solutions for the written questions. You are free to submit as many times as you like (which is useful
since you can see the autograder feedback immediately).
DO NOT change the name of any of the files or folders in the submission template. In other words,
your submitted files should have exactly the same names as those in the submission template. Do not
modify the directory structure.
Problem 1: Implementing Naive Bayes
In this question you will implement a Naive Bayes classifier for a text classification problem. You will be
given a collection of text articles, each coming from either the serious European magazine The Economist, or
from the not-so-serious American magazine The Onion. The goal is to learn a classifier that can distinguish
between articles from each magazine.
We have pre-processed the articles so that they are easier to use in your experiments. We extracted the
set of all words that occur in any of the articles. This set is called the vocabulary and we let V be the number
of words in the vocabulary. For each article, we produced a feature vector X = hX1 , . . . , XV i, where Xi is
equal to 1 if the ith word appears in the article and 0 otherwise. Each article is also accompanied by a class
label of either 1 for The Economist or 2 for The Onion. Later in the question we give instructions for loading
this data into Octave.
When we apply the Naive Bayes classification algorithm, we make two assumptions about the data: first,
we assume that our data is drawn iid from a joint probability distribution over the possible feature vectors
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X and the corresponding class labels Y ; second, we assume for each pair of features Xi and Xj with i 6= j
that Xi is conditionally independent of Xj given the class label Y (this is the Naive Bayes assumption).
Under these assumptions, a natural classification rule is as follows: Given a new input X, predict the most
probable class label Ŷ given X. Formally,
Ŷ = argmax P (Y = y|X).
y
Using Bayes Rule and the Naive Bayes assumption, we can rewrite this classification rule as follows:
P (X|Y = y)P (Y = y)
P (X)
(Bayes Rule)
= argmax P (X|Y = y)P (Y = y)
(Denominator does not depend on y)
Ŷ = argmax
y
y
= argmax P (X1 , . . . , XV |Y = y)P (Y = y)
y
Y
V
= argmax
P (Xw |Y = y) P (Y = y)
y
(Conditional independence).
w=1
The advantage of the Naive Bayes assumption is that it allows us to represent the distribution P (X|Y = y)
using many fewer parameters than would otherwise be possible. Specifically, since all the random variables
are binary, we only need one parameter to represent the distribution of Xw given Y for each w ∈ {1, . . . , V }
and y ∈ {1, 2}. This gives a total of 2V parameters. On the other hand, without the Naive Bayes assumption,
it is not possible to factor the probability as above, and therefore we need one parameter for all but one of
the 2V possible feature vectors X and each class label y ∈ {1, 2}. This gives a total of 2(2V − 1) parameters.
The vocabulary for our data has V ≈ 26, 000 words. Under the Naive Bayes assumption, we require on the
order of 52, 000 parameters, while without it we need more than 107000 !
Of course, since we don’t know the true joint distribution over feature vectors X and class labels Y , we
need to estimate the probabilities P (X|Y = y) and P (Y = y) from the training data. For each word index
w ∈ {1, . . . , V } and class label y ∈ {1, 2}, the distribution of Xw given Y = y is a Bernoulli distribution with
parameter θyw . In other words, there is some unknown number θyw such that
P (Xw = 1|Y = y) = θyw
and P (Xw = 0|Y = y) = 1 − θyw .
We believe that there is a non-zero (but maybe very small) probability that any word in the vocabulary can
appear in an article from either The Onion or The Economist. To make sure that our estimated probabilities
are always non-zero, we will impose a Beta(2,1) prior on θyw and compute the MAP estimate from the
training data.
Similarly, the distribution of Y (when we consider it alone) is a Bernoulli distribution (except taking
values 1 and 2 instead of 0 and 1) with parameter ρ. In other words, there is some unknown number ρ such
that
P (Y = 1) = ρ and P (Y = 2) = 1 − ρ.
In this case, since we have many examples of articles from both The Economist and The Onion, there is no
risk of having zero-probability estimates, so we will instead use the MLE.
Programming Instructions
Parts (a) through (e) of this question each ask you to implement one function related to the Naive Bayes
classifier. You will submit your code online through the CMU autolab system, which will execute it remotely
against a suite of tests. Your grade will be automatically determined from the testing results. Since you get
immediate feedback after submitting your code and you are allowed to submit as many different versions as
you like (without any penalty), it is easy for you to check your code as you go.
Our autograder requires that you write your code in Octave. Octave is a free scientific programming
language with syntax identical to that of MATLAB. Installation instructions can be found on the Octave
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website (http://www.gnu.org/software/octave/), and we have posted links to several Octave and MATLAB tutorials on Piazza.
To get started, you can log into the autolab website (https://autolab.cs.cmu.edu). From there you
should see 10-601B in your list of courses. Download the template for Homework 3 and extract the contents
(i.e., by executing tar xvf hw3.tar at the command line). In the archive you will find one .m file for
each of the functions that you are asked to implement and a file that contains the data for this problem,
HW3Data.mat. To finish each programming part of this problem, open the corresponding .m file and complete
the function defined in that file. When you are ready to submit your solutions, you will create a new tar
archive of the top-level directory (i.e., by executing tar cvf hw3.tar hw3) and upload that through the
Autolab website.
The file HW3Data.mat contains the data that you will use in this problem. You can load it from Octave by
executing load("HW3Data.mat") in the Octave interpreter. After loading the data, you will see that there
are 7 variables: Vocabulary, XTrain, yTrain, XTest, yTest, XTrainSmall, and yTrainSmall.
• Vocabulary is a V ×1 dimensional cell array that that contains every word appearing in the documents.
When we refer to the j th word, we mean Vocabulary(j,1).
• XTrain is a n × V dimensional matrix describing the n documents used for training your Naive Bayes
classifier. The entry XTrain(i,j) is 1 if word j appears in the ith training document and 0 otherwise.
• yTrain is a n×1 dimensional matrix containing the class labels for the training documents. yTrain(i,1)
is 1 if the ith document belongs to The Economist and 2 if it belongs to The Onion.
• XTest and yTest are the same as XTrain and yTrain, except instead of having n rows, they have m
rows. This is the data you will test your classifier on and it should not be used for training.
• Finally, XTrainSmall and yTrainSmall are subsets of XTrain and yTrain which are used in the final
question.
Logspace Arithmetic
When working with very large or very small numbers (such as probabilities), it is useful to work in logspace to
avoid numerical precision issues. In logspace, we keep track of the logs of numbers, instead of the numbers
themselves. For example, if p(x) and p(y) are probability values, instead of storing p(x) and p(y) and
computing p(x) ∗ p(y), we work in log space by storing log(p(x)), log(p(y)), and we can compute the log of
the product, log(p(x) ∗ p(y)) by taking the sum: log(p(x) ∗ p(y)) = log(p(x)) + log(p(y)).
(a) [1 Point] Complete the function logProd(x) which takes as input a vector of numbers in logspace
Q (i.e.,
xi = log pi ) and returns the product of those numbers in logspace—i.e., logProd(x) = log i pi .
Training Naive Bayes
(b) [4 Points] Complete the function [D] = NB XGivenY(XTrain, yTrain). The output D is a 2 × V
matrix, where for any word index w ∈ {1, . . . , V } and class index y ∈ {1, 2}, the entry D(y,w) is the
MAP estimate of θyw = P (Xw = 1|Y = y) with a Beta(2,1) prior distribution.
(c) [4 Points] Complete the function [p] = NB YPrior(yTrain). The output p is the MLE for ρ =
P (Y = 1).
(d) [8 Points] Complete the function [yHat] = NB Classify(D, p, X). The input X is an m × V matrix
containing m feature vectors (stored as its rows). The output yHat is a m × 1 vector of predicted class
labels, where yHat(i) is the predicted label for the ith row of X. [Hint: In this function, you will want
to use the logProd function to avoid numerical problems.]
(e) [1 Point] Complete the function [error] = ClassificationError(yHat, yTruth), which takes two
vectors of equal length and returns the proportion of entries that they disagree on.
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Questions
(g) [4 Points] Train your classifier on the data contained in XTrain and yTrain by running
D = NB_XGivenY(XTrain, yTrain);
p = NB_YPrior(yTrain);
Use the learned classifier to predict the labels for the article feature vectors in XTrain and XTest by
running
yHatTrain = NB_Classify(D, p, XTrain);
yHatTest = NB_Classify(D, p, XTest);
Use the function ClassificationError to measure and report the training and testing error by running
trainError = ClassificationError(yHatTrain, yTrain);
testError = ClassificationError(yHatTest, yTest);
How do the train and test errors compare? Explain any significant differences.
(h) [4 Points] Repeat the steps from part (g), but this time use the smaller training set XTrainSmall and
yTrainSmall. Explain any difference between the train and test error in this question and in part (g).
[Hint: When we have less training data, does the prior have more or less impact on our classifier?]
(i) [4 Points] Finally, we will try to interpret the learned parameters. Train your classifier on the data
contained in XTrain and yTrain. For each class label y ∈ {1, 2}, list the five words that the model says
are most likely to occur in a document from class y. Also for each class label y ∈ {1, 2}, list the five
words w that maximize the following quantity:
P (Xw = 1|Y = y)
.
P (Xw = 1|Y 6= y)
Which list of words describes the two classes better? Briefly explain your reasoning. (Note that some
of the words may look a little strange because we have run them through a stemming algorithm that
tries to make words with common roots look the same. For example, “stemming” and “stemmed” would
both become “stem”.)
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