# Homework

## Spring 2012

No late homework!

Lecture notes are graded as a homework assignment and are due by the beginning of the next lecture.

Homework 7
Due Wed 4/18 5pm

Implement EM to train an HMM for the data from Homework 6. The model should have four hidden states with gaussian observation probabilities. Does the HMM model the data better than the original mixture of gaussians?

Homework 6
Due Wed 3/7 5pm

Implement EM fitting of a mixture of gaussians on the two-dimensional data set points.dat. You should try different numbers of mixtures, as well as tied vs. separate covariance matrices for each gaussian. Which model seems to fit the data best?

Homework 5
Due Tu 2/21 in class

• Bishop 6.2, 6.8, 6.9
• Bishop 7.2, 7.7
Homework 4
Due Web 2/15 5pm by turn_in script

• Implement the raw (no learning rate) and gradient descent versions of the perceptron algorithm for the voting data. Experiment with different learning rate schedules. How do the results compare with naive bayes?
Homework 3
Due Web 2/1 5pm by turn_in script
• Implement naive bayes on the voting dataset using MATLAB. Plot results on dev and test sets as a function of the number of features used.
Homework 2
Due Thursday 1/26 in class

• Given a set of n i.i.d. observations x_1 to x_n, let P be the empirical distribution P(k)=c(k)/n. Prove the the distribution Q that minimizes the KL divergence with P, D(P||Q), also maximizes the probability of the data Q(x_1, ..., x_n).
• Prove (step by step) that entropy for a discrete random variable is maximized by the uniform distribution.
• Prove that if X and Y are independent, Var[X+Y] = Var[X] + Var[Y].
Homework 1
Due Tuesday 1/24 in class

Bishop ex. 1.3, 1.11

gildea @ cs rochester edu
April 12, 2012