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\lecture{Artificial Intelligence}{HW6: Probability}{CS5300, Spring 2009}

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% "CS5300, Spring 2009" WITH YOUR NAME AND UID.

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\section{Probability and Hamburgers}

Doctors have found that people with Kreuzfeld-Jacob disease (KJ) are
almost invariably ate lost of hamburgers, thus
$p(\text{HamburgerEater}|\text{KJ}) = 0.9$.  KJ is a rare disease:
about $1$ in $100,000$ people get it.  Eating hamburgers is
widespread: $p(\text{HamburgerEater}) = 0.5$.  What is the probability
that a regular hamburger eater will have KJ disease?  Show your work.

\section{Modeling Independence}

Consider the following set up.  A student is taking a class.  The
student might Study or not.  The student might Know-the-material or
not.  The student might Pass or not.  For each of the following
independence/conditional independence claims, state whether or not you
think it is true and provide a one-sentence justification.

\begin{enumerate}
\item $S \indep K$
\item $S \indep P$
\item $K \indep P$
\item $P \indep S \| K$
\item $P \indep K \| S$
\item $S \indep K \| P$
\end{enumerate}

\section{Independence in Joint Probability Tables}

Suppose we have a three variable model with variables $A$, $B$ and
$C$.  We assume $p(A,B,C) = p(A) p(B \| A) p(C \| B)$.

\begin{enumerate}
\item What (conditional) independence assumptions are made by the
  assumption of how this distribution factorizes?

\item Write out the full joint distribution based on the conditional
  probability tables shown below:

\parbox[t]{.3\textwidth}{
$p(A)$:
\begin{tabular}{c|c}
$A$ & $p$ \\
\hline
T & $3/4$ \\
F & $1/4$
\end{tabular}
}\parbox[t]{.3\textwidth}{
$p(B\|A=T)$:
\begin{tabular}{c|c}
$B$ & $p$ \\
\hline
T & $1/2$ \\
F & $1/2$
\end{tabular}

$p(B\|A=F)$:
\begin{tabular}{c|c}
$B$ & $p$ \\
\hline
T & $2/3$ \\
F & $1/3$
\end{tabular}
}\parbox[t]{.3\textwidth}{
$p(C\|B=T)$:
\begin{tabular}{c|c}
$C$ & $p$ \\
\hline
T & $1/5$ \\
F & $4/5$
\end{tabular}

$p(C\|B=F)$:
\begin{tabular}{c|c}
$C$ & $p$ \\
\hline
T & $3/4$ \\
F & $1/4$
\end{tabular}
}

\end{enumerate}

\end{document}