\chapter{Probabilistic representation and reasoning (and burglars)}
\section{Bayesian network and the conditional probability tables}
\begin{figure}[H]
	\caption{Bayesian network, visual representation}
	\centering
	\includegraphics[scale=0.5]{d1.eps}
\end{figure}

We introduced a \textit{Noisy OR} to represent the causal independence of
\textit{Burglar} and \textit{Earthquake} on Alarm. Probabilities for the causes
of the alarm are calculated using days, in practice this means that the
smallest discrete time interval is one day. The calculation for the probability
of a burglar is then calculated with the following formula(taking leap years
into account and assuming a standard gregorian calendar).
$$\frac{1}{365 + 0.25 - 0.01 - 0.0025}=\frac{1}{365.2425}$$

This gives the following probability distributions\\
\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{Earthquake}\\
	\hline
	T & $0.0027$ & $0.9972$ \\
	F & $0.9973$ & $0.0027$\\
	\hline
\end{tabular}
%
\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{Burglar}\\
	\hline
	T & $0.0027$ & $0.9973$ \\
	F & $0.9973$ & $0.0027$\\
	\hline
\end{tabular}

\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{$I_1$}\\
	Earthquake & T & F\\
	\hline
	T & $0.2$ & $0.08$\\
	F & $0$ & $1$\\
	\hline
\end{tabular}
\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{$I_2$}\\
	Burglar & T & F\\
	\hline
	T & $0.95$ & $0.05$\\
	F & $0$ & $1$\\
	\hline
\end{tabular}
\begin{tabular}{|ll|ll|}
	\hline
	&& \multicolumn{2}{c|}{Alarm}\\
	$I_1$ & $I_2$ & T & F\\
	\hline
	T & T & $1$ & $0$\\
	T & F & $1$ & $0$\\
	F & T & $1$ & $0$\\
	F & F & $0$ & $1$\\
	\hline
\end{tabular}

\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{Watson}\\
	Alarm & T & F\\
	\hline
	T & $0.8$ & $0.2$\\
	F & $0.4$ & $0.6$\\
	\hline
\end{tabular}
\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{Gibbons}\\
	Alarm & T & F\\
	\hline
	T & $0.99$ & $0.01$\\
	F & $0.04$ & $0.96$\\
	\hline
\end{tabular}
\begin{tabular}{|l|ll|}
	\hline
	& \multicolumn{2}{c|}{Radio}\\
	Earthquake & T & F\\
	\hline
	T & $0.9998$ & $0.0002$\\
	F & $0.0002$ & $0.9998$\\
	\hline
\end{tabular}
