\chapter{Probabilistic representation and reasoning (and burglars)}
\section{Formal description}
In our representation of the model we chose to introduce a \textit{Noisy OR} to
represent the causal independence of \textit{Burglar} and \textit{Earthquake}
on \textit{Alarm}. The visual representation of the network is visible in
Figure~\ref{bnetwork21}

\begin{figure}[H]
	\caption{Bayesian network, visual representation}
	\label{bnetwork21}
	\centering
	\includegraphics[scale=0.5]{d1.eps}
\end{figure}

As for the probabilities for \textit{Burglar} and \textit{Earthquake} we chose
to calculate them using days the unit. Calculation for the probability of a
\textit{Burglar} event happening at some day is then this(assuming a gregorian
calendar and leap days).
$$\frac{1}{365 + 0.25 - 0.01 - 0.0025}=\frac{1}{365.2425}$$

This gives the following probability distributions visible in
Table~\ref{probdist}

\begin{table}[H]
	\label{probdist}
	\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 & $0.2$ & $0.8$\\
		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}
\end{table}

\section{Implementation}
This distribution results in the \textit{AILog} code in Listing~\ref{alarm.ail}

\begin{listing}
	\label{alarm.ail}
	\caption{Alarm.ail}
	\inputminted[linenos,fontsize=\footnotesize]{prolog}{./src/alarm.ail}
\end{listing}
