report/ass2-1.tex

   1 \chapter{Probabilistic representation and reasoning (and burglars)}
   2 \section{Formal description}
   3 In our representation of the model we chose to introduce a \textit{Noisy OR} to
   4 represent the causal independence of \textit{Burglar} and \textit{Earthquake}
   5 on \textit{Alarm}. The visual representation of the network is visible in
   6 Figure~\ref{bnetwork21}
   7
   8 \begin{figure}[H]
   9         \caption{Bayesian network, visual representation}
  10         \label{bnetwork21}
  11         \centering
  12         \includegraphics[scale=0.5]{d1.eps}
  13 \end{figure}
  14
  15 As for the probabilities for \textit{Burglar} and \textit{Earthquake} we chose
  16 to calculate them using days the unit. Calculation for the probability of a
  17 \textit{Burglar} event happening at some day is then this(assuming a gregorian
  18 calendar and leap days).
  19 $$\frac{1}{365 + 0.25 - 0.01 - 0.0025}=\frac{1}{365.2425}$$
  20
  21 This gives the following probability distributions visible in
  22 Table~\ref{probdist}
  23
  24 \begin{table}[H]
  25         \label{probdist}
  26         \begin{tabular}{|l|l|}
  27                 \hline
  28                 & Earthquake\\
  29                 \hline
  30                 T & $0.0027$\\
  31                 F & $0.9973$\\
  32                 \hline
  33         \end{tabular}
  34         %
  35         \begin{tabular}{|l|l|}
  36                 \hline
  37                 & Burglar\\
  38                 \hline
  39                 T & $0.0027$\\
  40                 F & $0.9973$\\
  41                 \hline
  42         \end{tabular}
  43
  44         \begin{tabular}{|l|ll|}
  45                 \hline
  46                 & \multicolumn{2}{c|}{$I_1$}\\
  47                 Earthquake & T & F\\
  48                 \hline
  49                 T & $0.2$ & $0.8$\\
  50                 F & $0$ & $1$\\
  51                 \hline
  52         \end{tabular}
  53         \begin{tabular}{|l|ll|}
  54                 \hline
  55                 & \multicolumn{2}{c|}{$I_2$}\\
  56                 Burglar & T & F\\
  57                 \hline
  58                 T & $0.95$ & $0.05$\\
  59                 F & $0$ & $1$\\
  60                 \hline
  61         \end{tabular}
  62         \begin{tabular}{|ll|ll|}
  63                 \hline
  64                 && \multicolumn{2}{c|}{Alarm}\\
  65                 $I_1$ & $I_2$ & T & F\\
  66                 \hline
  67                 T & T & $1$ & $0$\\
  68                 T & F & $1$ & $0$\\
  69                 F & T & $1$ & $0$\\
  70                 F & F & $0$ & $1$\\
  71                 \hline
  72         \end{tabular}
  73
  74         \begin{tabular}{|l|ll|}
  75                 \hline
  76                 & \multicolumn{2}{c|}{Watson}\\
  77                 Alarm & T & F\\
  78                 \hline
  79                 T & $0.8$ & $0.2$\\
  80                 F & $0.4$ & $0.6$\\
  81                 \hline
  82         \end{tabular}
  83         \begin{tabular}{|l|ll|}
  84                 \hline
  85                 & \multicolumn{2}{c|}{Gibbons}\\
  86                 Alarm & T & F\\
  87                 \hline
  88                 T & $0.99$ & $0.01$\\
  89                 F & $0.04$ & $0.96$\\
  90                 \hline
  91         \end{tabular}
  92         \begin{tabular}{|l|ll|}
  93                 \hline
  94                 & \multicolumn{2}{c|}{Radio}\\
  95                 Earthquake & T & F\\
  96                 \hline
  97                 T & $0.9998$ & $0.0002$\\
  98                 F & $0.0002$ & $0.9998$\\
  99                 \hline
 100         \end{tabular}
 101 \end{table}
 102
 103 \section{Implementation}
 104 This distribution results in the \textit{AILog} code in Listing~\ref{alarm.ail}
 105
 106 \begin{listing}[H]
 107         \label{alarm.ail}
 108         \caption{alarm.ail}
 109         \inputminted[linenos,fontsize=\footnotesize]{prolog}{./src/alarm.ail}
 110 \end{listing}
 111
 112 \section{Queries}
 113 Using the following queries the probabilities or as follows:\\
 114 \begin{enumerate}[a)]
 115         \item $P(\text{Burglary})=
 116                 0.002737757092501968$
 117         \item $P(\text{Burglary}|\text{Watson called})=
 118                 0.005321803679438259$
 119         \item $P(\text{Burglary}|\text{Watson called}\wedge\text{Gibbons called})=
 120                 0.11180941544755249$
 121         \item $P(\text{Burglary}|\text{Watson called}\wedge\text{Gibbons called}
 122                 \wedge\text{Radio})=0.01179672476662423$
 123 \end{enumerate}
 124
 125 \begin{listing}[H]
 126         \begin{minted}[fontsize=\footnotesize]{prolog}
 127 ailog: predict burglar.
 128 Answer: P(burglar|Obs)=0.002737757092501968.
 129   [ok,more,explanations,worlds,help]: ok.
 130
 131 ailog: observe watson.
 132 Answer: P(watson|Obs)=0.4012587986186947.
 133   [ok,more,explanations,worlds,help]: ok.
 134
 135 ailog: predict burglar.
 136 Answer: P(burglar|Obs)=[0.005321803679438259,0.005321953115441623].
 137   [ok,more,explanations,worlds,help]: ok.
 138
 139 ailog: observe gibbons.
 140 Answer: P(gibbons|Obs)=[0.04596053565368094,0.045962328885721306].
 141   [ok,more,explanations,worlds,help]: ok.
 142
 143 ailog: predict burglar.
 144 Answer: P(burglar|Obs)=[0.11180941544755249,0.1118516494624678].
 145   [ok,more,explanations,worlds,help]: ok.
 146
 147 ailog: observe radio.
 148 Answer: P(radio|Obs)=[0.02582105837443645,0.025915745316785182].
 149   [ok,more,explanations,worlds,help]: ok.
 150
 151 ailog: predict burglar.
 152 Answer: P(burglar|Obs)=[0.01179672476662423,0.015584580594335082].
 153   [ok,more,explanations,worlds,help]: ok.
 154         \end{minted}
 155 \end{listing}
 156
 157 \section{Comparison with manual calculation}
 158 Querying the \textit{Alarm} variable gives the following answer
 159 \begin{minted}{prolog}
 160         ailog: predict alarm.
 161         Answer: P(alarm|Obs)=0.0031469965467367292.
 162           [ok,more,explanations,worlds,help]: ok.
 163 \end{minted}
 164
 165 Using formula: $P(i_1|C_1)+P(i_2|C_2)(1-P(i_1|C_1))$ we can calculate the
 166 probability of the \textit{Alarm} variable using variable elimination. This
 167 results in the following answer:
 168 $$0.2*0.0027+0.95*0.0027*(1-0.2*0.0027)=0.00314699654673672941001347$$
 169
 170 There is a slight difference in probability. This is probably due to the
 171 precision of the \textit{AILog} module. The manual calculation was done with
 172 arbitrary precision. Manual calculation takes a lot longer and therefore one
 173 can prefer the \textit{AILog} method when speed is of an essence. When
 174 precision is necessary manual calculation is preferred.
 175
 176 \section{Burglary problem with extended information}
 177