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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!

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

Consider the graphical model (shown above) over five binary random variables:

- [20 points] List the independence assumptions captured in the model between the following pairs of random 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.
- $L$ and $R$
- $T$ and $R$

- [20 points] More independence:
- Assuming that the value of $L$ 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. - 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.

- [10 points] Factor the joint distribution represented by the entire model shown in the figure according to the explicit independence assumptions represented in the model.
- [10 points] Write the four entries and their values in the conditional distribution for $P(L=0 | M=m,S=s)$ (for $m \in \{0,1\}$ and $s \in \{0,1\}$).
- [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.
- [10 points] Compute $P(T=1, R=0, L=0)$. Show your work.
- [10 points] Compute $P(T=1 | R=0, L=0)$. Show your work.
- [10 points] Prove that the relationship we call conditional independence is symmetric. Apply the same standard of proof as in homework 0.1.
- In other words, show that (a) $P(X | Y, Z) = P(X | Z)$ if and only if $P(Y | X, Z) = P(Y | Z)$
- Equivalently, show that (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).
- To be clear, prove (a) or (b).

Your **typed** report should include:

- Your work and solutions for the exercises

Organize your report and use clear headings and explanations where needed.

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