# 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!

## Preparation

In addition to the material in recent lectures, consider a brief tutorial on the topic of influence in Bayes nets.

## Exercises

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

### Question 1

[10 points] Consider the graphical model (shown above) over five binary random variables. Factor the joint distribution represented by the entire model shown in the figure according to the explicit independence assumptions represented in the model.

### Question 2

[20 points] List the independence relations captured in the model (in the above figure) between the following pairs of random variables. Be sure to consider all of the cases in which the (three) other variables in the model have known values and when they do not.

• $L$ and $R$
• To clarify: You must answer whether $L \perp R | X$ for $X \in 2^{\{S,M,T\}}$ (that's 8 answers in all).
• In other words, indicate whether $R$ influences $L$ in the presence of the observation $X \in 2^{\{S,M,T\}}$.
• Note that you don't have to say whether $L$ influences $R$ because of the symmetry of conditional independence.
• $T$ and $R$
• As in the first bullet, there are 8 answers to be given here as well.

2015: Note: Assume that the CPTs in the model contain values such that if influence is possible between a pair of nodes – given values of other variables in the model – using the heuristics we have discussed, then conditional independence will not hold. Under that assumption, the structure of the graph will be sufficient to tell you that a given conditional independence statement is false.

### Question 3

[24 points] More independence:

• Assuming that the value of $L$ is known, list all independence relations between $T$ and (each of the three) other variables. Be sure to consider all of the cases in which the other (i.e., the remaining two) variables in the model have known values and when they do not.
• To clarify: You must answer whether $T \perp X | L, Y$, where $X \in \{S, M, R\}$ and $Y \in 2^{\{S, M, R\}-X}$, (that's 12 answers in all).
• In other words, indicate whether $X \in \{S, M, R\}$ influences $T$ in the presence of the observation $L$ and the observation $Y \in 2^{\{S, M, R\}-X}$.
• Note that you don't have to say whether $T$ influences $X$ because of the symmetry of conditional independence.
• Assuming that the value of $S$ is known, list all independence relations between $T$ and other variables. Be sure to consider all of the cases in which the other variables in the model have known values and when they do not.
• As in the first bullet, there are 12 answers to be given here as well.

2015: Note: Assume that the CPTs in the model contain values such that if influence is possible between a pair of nodes – given values of other variables in the model – using the heuristics we have discussed, then conditional independence will not hold. Under that assumption, the structure of the graph will be sufficient to tell you that a given conditional independence statement is false.

### Question 4

[10 points] Write the four probabilities $P(L=0 | M=m,S=s)$ (for $m \in \{0,1\}$ and $s \in \{0,1\}$), both with the symbols and the resulting values.

### Question 5

[10 points]

• (a) Write an expression for the joint probability $P(T=1, R=0, L=0, M=0, S=1)$ in terms of the probabilities given in the model (use the symbolic forms).
• (b) Then compute the actual probability.

### Question 6

[16 points] Compute $P(T=1, R=0, L=0)$. Show your work.

### Question 7

[10 points] Compute $P(T=1 | R=0, L=0)$. Show your work.