report/ass2-1.tex

   1 \chapter{Probabilistic representation and reasoning (and burglars)}
   2 \section{Bayesian network and the conditional probability tables}
   3 \begin{figure}[H]
   4         \caption{Bayesian network, visual representation}
   5         \centering
   6         \includegraphics[scale=0.5]{d1.eps}
   7 \end{figure}
   8
   9 We introduced a \textit{Noisy OR} to represent the causal independence of
  10 \textit{Burglar} and \textit{Earthquake} on Alarm. Probabilities for the causes
  11 of the alarm are calculated using days, in practice this means that the
  12 smallest discrete time interval is one day. The calculation for the probability
  13 of a burglar is then calculated with the following formula(taking leap years
  14 into account and assuming a standard gregorian calendar).
  15 $$\frac{1}{365 + 0.25 - 0.01 - 0.0025}=\frac{1}{365.2425}$$
  16
  17 This gives the following probability distributions\\
  18 \begin{tabular}{|l|ll|}
  19         \hline
  20         & \multicolumn{2}{c|}{Earthquake}\\
  21         \hline
  22         T & $0.0027$ & $0.9972$ \\
  23         F & $0.9973$ & $0.0027$\\
  24         \hline
  25 \end{tabular}
  26 %
  27 \begin{tabular}{|l|ll|}
  28         \hline
  29         & \multicolumn{2}{c|}{Burglar}\\
  30         \hline
  31         T & $0.0027$ & $0.9973$ \\
  32         F & $0.9973$ & $0.0027$\\
  33         \hline
  34 \end{tabular}
  35
  36 \begin{tabular}{|l|ll|}
  37         \hline
  38         & \multicolumn{2}{c|}{$I_1$}\\
  39         Earthquake & T & F\\
  40         \hline
  41         T & $0.2$ & $0.8$\\
  42         F & $0$ & $1$\\
  43         \hline
  44 \end{tabular}
  45 \begin{tabular}{|l|ll|}
  46         \hline
  47         & \multicolumn{2}{c|}{$I_2$}\\
  48         Burglar & T & F\\
  49         \hline
  50         T & $0.95$ & $0.05$\\
  51         F & $0$ & $1$\\
  52         \hline
  53 \end{tabular}
  54 \begin{tabular}{|ll|ll|}
  55         \hline
  56         && \multicolumn{2}{c|}{Alarm}\\
  57         $I_1$ & $I_2$ & T & F\\
  58         \hline
  59         T & T & $1$ & $0$\\
  60         T & F & $1$ & $0$\\
  61         F & T & $1$ & $0$\\
  62         F & F & $0$ & $1$\\
  63         \hline
  64 \end{tabular}
  65
  66 \begin{tabular}{|l|ll|}
  67         \hline
  68         & \multicolumn{2}{c|}{Watson}\\
  69         Alarm & T & F\\
  70         \hline
  71         T & $0.8$ & $0.2$\\
  72         F & $0.4$ & $0.6$\\
  73         \hline
  74 \end{tabular}
  75 \begin{tabular}{|l|ll|}
  76         \hline
  77         & \multicolumn{2}{c|}{Gibbons}\\
  78         Alarm & T & F\\
  79         \hline
  80         T & $0.99$ & $0.01$\\
  81         F & $0.04$ & $0.96$\\
  82         \hline
  83 \end{tabular}
  84 \begin{tabular}{|l|ll|}
  85         \hline
  86         & \multicolumn{2}{c|}{Radio}\\
  87         Earthquake & T & F\\
  88         \hline
  89         T & $0.9998$ & $0.0002$\\
  90         F & $0.0002$ & $0.9998$\\
  91         \hline
  92 \end{tabular}
  93
  94 \paragraph{Implementation}\strut\\
  95 This distribution results in the \textit{AILog} code in Listing~\ref{alarm.ail}
  96
  97 \begin{listing}
  98         \label{alarm.ail}
  99         \caption{Alarm.ail}
 100         \inputminted[linenos,fontsize=\footnotesize]{prolog}{./src/alarm.ail}
 101 \end{listing}