• No extension will be provided, unless for serious documented reasons.
• Start early!
• Study the material taught in class, and feel free to do so in small groups, but the solutions should be a product of your own work.
• This is not a multiple choice homework; reasoning, and mathematical proofs are required before giving your final answer.
1 Probability [45 points]
Solve the following problems:
a. (5pts) Give an example of a random variable for which Chebyshev’s inequality is tight, namely the inequality holds as equality.
b. (10pts) You wish to send a single bit b from Boston (place A1) to San Francisco (place An) through a chain A1 → A2 → ... → An of intermediate place. Sending the bit b from one place to another place flips its value with probability p. What is the probability that San Francisco will receive the right value b instead of the wrong value 1 − b?
c. (15pts) A,B tell the truth with probability p and lie with probability 1 − p. A makes a statement and B confirms the statement by A is true. What is the probability that A is actually telling the truth?
d. (15pts) Let X,Y,Z be uniform random variables in [0,1]. Compute the probability
P(X + Y + Z ≤ 1).
2 Spam or Ham: Naive Bayes Classifier [55 Points]
Overview/Task
1. Download the code template notebook HW3-coding.ipynb, the training file TRAIN balanced ham spam.csv, and the test file TEST balanced ham spam.csv.
2. Do not change any function names nor variable names that are outside of coding prompts.
For instance, in the image above, you should NOT edit the name nor the parameters of the function “prior”, the variable names “ham prior” and “spam prior”, and the return variables of the same name
Recommended task order
In order to provide some guidance, I am giving the following order/checklist to solve this task:
1. 10 points: Compute the prior of whether an email is spam or ham from your training data.
2. 15 points: Compute the likelihood.
(a) For the computation of the likelihood and prior, please refer to slide 15, lect. 5.
3. 25 points: Implement Bayes classifier. Specifically, write code that uses the prior and the likelihood to maximize the posterior. Use this to make a decision on whether or not a given email is spam or ham.
4. 5 points: Evaluate your prediction by computing the accuracy, precision, and recall (WITHOUT using external libraries such as sklearn).
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