**This is an old revision of the document!**

# Bayes Nets

## Objectives

This assignment is designed to:

• give you practice thinking about probabilistic models in the form of Bayes Nets (directed graphical models)
• help you become more fluent with the terminology and the techniques of the course
• prepare you to work with interesting graphical models of natural phenomena!

## Exercises

Show your work. Be clear and concise. This assignment must be typed.

1. [10 points] Prove that the relationship we call conditional independence is symmetric. In other words, Prove either (a) or (b) (since they are equivalent), and apply the same standard of proof as in assignment 1:
• (a) $P(X | Y, Z) = P(X | Z)$ if and only if $P(Y | X, Z) = P(Y | Z)$
• (b) $P(X, Y | Z) = P(X | Z) \cdot P(Y|Z)$ if and only if $P(Y, X | Z) = P(Y | Z) \cdot P(X | Z)$
• (in other words, the “given $Z$” stays the same, while $X$ and $Y$ trade places).
2. [20 points: 10 points each] (based on exercise 2.2 in Koller and Friedman) Independence:
• Prove that for binary random variables $X$ and $Y$, the event-level independence $(x^0 \bot y^0)$ implies random-variable independence $(X \bot Y)$. Use the usual standard of proof.
• Give a counterexample for nonbinary variables.
3. [20 points] Consider how to sample from a categorical distribution over four colors. Think of a spinner with four regions having probabilities $p_{red}$, $p_{green}$, $p_{yellow}$, and $p_{blue}$. Write pseudo-code for choosing a sample from this distribution.
4. [10 points] Does your pseudo-code scale to a distribution over ten thousand values? If not, rewrite it. If so, say why.
5. [20 points] Implement your pseudo-code, run it 100 times, and give the results as a vector of counts over the four colors.
6. [10 points] Normalize your count vector by 100. How does the result compare with your chosen parameters?
7. [10 points] Compute the mean and variance of the estimated multinomial distribution you just discovered.