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\lecture{Artificial Intelligence}{HW7: Bayes Nets}{CS5300, Spring 2009}

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\section{Independences from Probability Tables}

Consider the following joint probability table over three variables.
What independences and conditional independences can you find in this
distribution?  That is, if you factorize the joint probability as a
product of conditional probabilities, how small can you get the
conditional probability tables?  Write them out.

\begin{tabular}{|ccc|c|}
\hline
{\bf A} & {\bf B} & {\bf C} & $p$ \\
\hline
T & T & T & $1/16$ \\
T & T & F & $1/3$ \\
T & F & T & $1/32$ \\
T & F & F & $1/12$ \\
F & T & T & $3/16$ \\
F & T & F & $1/6$ \\
F & F & T & $3/32$ \\
F & F & F & $1/24$ \\
\hline
\end{tabular}

\section{Independence in Graphical Models}

Consider the graphical model shown below:

\includegraphics[width=0.5\textwidth]{hw7model}

Please answer the following conditional independence questions from
this model:

\begin{enumerate}
\item $A \indep H$
\item $A \indep H \| C$
\item $A \indep H \| C,F$
\item $E \indep B \| A$
\item $E \indep B \| C,F$
\item $E \indep B \| A,C,F$
\end{enumerate}

\newpage
\section{Inference by Enumeration and Variable Elimination}

Consider the graphical model for the alarm network (slide 9 of
day17, for instance).  Using inference by enumeration, compute the
following probabilities (show your work!!!):

\begin{enumerate}
\item $p(b, \lnot e \| a, j, m)$
\item $p(b \| a)$
\item $p(b \| e,a)$  (how does this compare to the previous one?)
\item $p(a \| j,\lnot m)$
\end{enumerate}

Now, repeat items (2) and (4) using variable elimination.  When you
have to choose a variable to eliminate, choose alphabetically.


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