\chapter{Probabilistic representation and reasoning (and burglars)}
-\section{Bayesian network and the conditional probability tables}
+\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}
-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).
+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\\
-\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}
+This gives the following probability distributions visible in
+Table~\ref{probdist}
-\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}
+\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}
-\paragraph{Implementation}\strut\\
+\section{Implementation}
This distribution results in the \textit{AILog} code in Listing~\ref{alarm.ail}
\begin{listing}