Description
1 Maximum Likelihood Estimation [15 pts]
Suppose we observe the values of n independent random variables X1, . . . , Xn drawn from the same
Bernoulli distribution with parameter θ
1
. In other words, for each Xi
, we know that
P(Xi = 1) = θ and P(Xi = 0) = 1 − θ.
Our goal is to estimate the value of θ from these observed values of X1 through Xn.
For any hypothetical value ˆθ, we can compute the probability of observing the outcome X1, . . . , Xn
if the true parameter value θ were equal to ˆθ. This probability of the observed data is often called
the likelihood, and the function L(θ) that maps each θ to the corresponding likelihood is called the
likelihood function. A natural way to estimate the unknown parameter θ is to choose the θ that
maximizes the likelihood function. Formally,
ˆθMLE = arg max
θ
L(θ).
(a) Write a formula for the likelihood function, L(θ) = P(X1, . . . , Xn; θ). Your function should
depend on the random variables X1, . . . , Xn and the hypothetical parameter θ. Does the
likelihood function depend on the order in which the random variables are observed ?
(b) Since the log function is increasing, the θ that maximizes the log likelihood `(θ) = log(L(θ)) is
the same as the θ that maximizes the likelihood. Find `(θ) and its first and second derivatives,
and use these to find a closed-form formula for the MLE.
(c) Suppose that n = 10 and the data set contains six 1s and four 0s. Write a short program likelihood.py that plots the likelihood function of this data for each value of ˆθ in
{0, 0.01, 0.02, . . . , 1.0} (use np.linspace(…) to generate this spacing). For the plot, the
x-axis should be θ and the y-axis L(θ). Scale your y-axis so that you can see some variation in
its value. Include the plot in your writeup (there is no need to submit your code). Estimate
ˆθMLE by marking on the x-axis the value of ˆθ that maximizes the likelihood. Does the answer
agree with the closed form answer ?
(d) Create three more likelihood plots: one where n = 5 and the data set contains three 1s and
two 0s; one where n = 100 and the data set contains sixty 1s and forty 0s; and one where
n = 10 and there are five 1s and five 0s. Include these plots in your writeup, and describe
how the likelihood functions and maximum likelihood estimates compare for the different
data sets.
2 Splitting Heuristic for Decision Trees [14 pts]
Recall that the ID3 algorithm iteratively grows a decision tree from the root downwards. On each
iteration, the algorithm replaces one leaf node with an internal node that splits the data based on
one decision attribute (or feature). In particular, the ID3 algorithm chooses the split that reduces
1This is a common assumption for sampling data. So we will denote this assumption as iid, short for Independent
and Identically Distributed, meaning that each random variable has the same distribution and is drawn independent
of all the other random variables
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the entropy the most, but there are other choices. For example, since our goal in the end is to have
the lowest error, why not instead choose the split that reduces error the most? In this problem, we
will explore one reason why reducing entropy is a better criterion.
Consider the following simple setting. Let us suppose each example is described by n boolean
features: X = hX1, . . . , Xni, where Xi ∈ {0, 1}, and where n ≥ 4. Furthermore, the target function
to be learned is f : X → Y , where Y = X1 ∨ X2 ∨ X3. That is, Y = 1 if X1 = 1 or X2 = 1
or X3 = 1, and Y = 0 otherwise. Suppose that your training data contains all of the 2n possible
examples, each labeled by f. For example, when n = 4, the data set would be
X1 X2 X3 X4 Y
0 0 0 0 0
1 0 0 0 1
0 1 0 0 1
1 1 0 0 1
0 0 1 0 1
1 0 1 0 1
0 1 1 0 1
1 1 1 0 1
X1 X2 X3 X4 Y
0 0 0 1 0
1 0 0 1 1
0 1 0 1 1
1 1 0 1 1
0 0 1 1 1
1 0 1 1 1
0 1 1 1 1
1 1 1 1 1
(a) How many mistakes does the best 1-leaf decision tree make over the 2n
training examples?
(The 1-leaf decision tree does not split the data even once. Make sure you answer for the
general case when n ≥ 4.)
(b) Is there a split that reduces the number of mistakes by at least one? (That is, is there a
decision tree with 1 internal node with fewer mistakes than your answer to part (a)?) Why
or why not?
(c) What is the entropy of the output label Y for the 1-leaf decision tree (no splits at all)?
(d) Is there a split that reduces the entropy of the output Y by a non-zero amount? If so, what
is it, and what is the resulting conditional entropy of Y given this split?
3 Entropy and Information [2 pts]
The entropy of a Bernoulli (Boolean 0/1) random variable X with p(X = 1) = q is given by
B(q) = −q log q − (1 − q) log(1 − q).
Suppose that a set S of examples contains p positive examples and n negative examples. The
entropy of S is defined as H(S) = B
p
p+n
.
(a) Based on an attribute Xj , we split our examples into k disjoint subsets Sk, with pk positive
and nk negative examples in each. If the ratio pk
pk+nk
is the same for all k, show that the
information gain of this attribute is 0.
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4 Programming exercise : Applying decision trees [24 pts]
Submission instructions
• Only provide answers and plots. Do not submit code.
Introduction2
The sinking of the RMS Titanic is one of the most infamous shipwrecks in history. On April 15,
1912, during her maiden voyage, the Titanic sank after colliding with an iceberg, killing 1502 out
of 2224 passengers and crew. This sensational tragedy shocked the international community and
led to better safety regulations for ships.
One of the reasons that the shipwreck led to such loss of life was that there were not enough lifeboats
for the passengers and crew. Although there was some element of luck involved in surviving the
sinking, some groups of people were more likely to survive than others, such as women, children,
and the upper-class.
In this problem, we ask you to complete the analysis of what sorts of people were likely to survive.
In particular, we ask you to apply the tools of machine learning to predict which passengers survived
the tragedy.
Starter Files
code and data
• code : titanic.py
• data : titanic_train.csv
documentation
• Decision Tree Classifier:
http://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.html
• Cross-Validation:
http://scikit-learn.org/stable/modules/generated/sklearn.cross_validation.train_test_split.html
• Metrics:
http://scikit-learn.org/stable/modules/generated/sklearn.metrics.accuracy_score.html
Download the code and data sets from the course website. For more information on the data set,
see the Kaggle description: https://www.kaggle.com/c/titanic/data. (The provided data sets
are modified versions of the data available from Kaggle.3
)
2This assignment is adapted from the Kaggle Titanic competition, available at https://www.kaggle.com/c/
titanic. Some parts of the problem are copied verbatim from Kaggle.
3Passengers with missing values for any feature have been removed. Also, the categorical feature Sex has been
mapped to {’female’: 0, ’male’: 1} and Embarked to {’C’: 0, ’Q’: 1, ’S’: 2}. If you are interested more
in this process of data munging, Kaggle has an excellent tutorial available at https://www.kaggle.com/c/titanic/
details/getting-started-with-python-ii.
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Note that any portions of the code that you must modify have been indicated with TODO. Do not
change any code outside of these blocks.
4.1 Visualization [4 pts]
One of the first things to do before trying any formal machine learning technique is to dive into
the data. This can include looking for funny values in the data, looking for outliers, looking at the
range of feature values, what features seem important, etc.
(a) Run the code (titanic.py) to make histograms for each feature, separating the examples
by class (e.g. survival). This should produce seven plots, one for each feature, and each plot
should have two overlapping histograms, with the color of the histogram indicating the class.
For each feature, what trends do you observe in the data?
4.2 Evaluation [20 pts]
Now, let us use scikit-learn to train a DecisionTreeClassifier on the data.
Using the predictive capabilities of the scikit-learn package is very simple. In fact, it can be
carried out in three simple steps: initializing the model, fitting it to the training data, and predicting
new values.4
(b) Before trying out any classifier, it is often useful to establish a baseline. We have implemented
one simple baseline classifier, MajorityVoteClassifier, that always predicts the majority
class from the training set. Read through the MajorityVoteClassifier and its usage and
make sure you understand how it works.
Your goal is to implement and evaluate another baseline classifier, RandomClassifier, that
predicts a target class according to the distribution of classes in the training data set. For
example, if 60% of the examples in the training set have Survived = 0 and 40% have
Survived = 1, then, when applied to a test set, RandomClassifier should randomly predict
60% of the examples as Survived = 0 and 40% as Survived = 1.
Implement the missing portions of RandomClassifier according to the provided specifications. Then train your RandomClassifier on the entire training data set, and evaluate its
training error. If you implemented everything correctly, you should have an error of 0.485.
(c) Now that we have a baseline, train and evaluate a DecisionTreeClassifier (using the class
from scikit-learn and referring to the documentation as needed). Make sure you initialize
your classifier with the appropriate parameters; in particular, use the ‘entropy’ criterion
discussed in class. What is the training error of this classifier?
(d) So far, we have looked only at training error, but as we learned in class, training error is a
poor metric for evaluating classifiers. Let us use cross-validation instead.
4Note that almost all of the model techniques in scikit-learn share a few common named functions, once
they are initialized. You can always find out more about them in the documentation for each model. These are
some-model-name.fit(…), some-model-name.predict(…), and some-model-name.score(…).
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Implement the missing portions of error(…) according to the provided specifications. You
may find it helpful to use train_test_split(…) from scikit-learn. To ensure that we
always get the same splits across different runs (and thus can compare the classifier results),
set the random_state parameter to be the trial number.
Next, use your error(…) function to evaluate the training error and (cross-validation) test
error of each of your three models. To do this, generate a random 80/20 split of the training
data, train each model on the 80% fraction, evaluate the error on either the 80% or the 20%
fraction, and repeat this 100 times to get an average result. What are the average training
and test error of each of your classifiers on the Titanic data set?
(e) One problem with decision trees is that they can overfit to training data, yielding complex
classifiers that do not generalize well to new data. Let us see whether this is the case for the
Titanic data.
One way to prevent decision trees from overfitting is to limit their depth. Repeat your crossvalidation experiments but for increasing depth limits, specifically, 1, 2, . . . , 20. Then plot the
average training error and test error against the depth limit. (Also plot the average test error
for your baseline classifiers. As the baseline classifiers are independent of the depth limit,
their plots should be flat lines.) Include this plot in your writeup, making sure to label all
axes and include a legend for your classifiers. What is the best depth limit to use for this
data? Do you see overfitting? Justify your answers using the plot.
(f) Another useful tool for evaluating classifiers is learning curves, which show how classifier
performance (e.g. error) relates to experience (e.g. amount of training data).
Run another experiment using a decision tree with the best depth limit you found above.
This time, vary the amount of training data by starting with splits of 0.05 (5% of the data
used for training) and working up to splits of size 0.95 (95% of the data used for training) in
increments of 0.05. Then plot the decision tree training and test error against the amount of
training data. (Also plot the average test error for your baseline classifiers.) Include this plot
in your writeup, and provide a 1-2 sentence description of your observations.
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