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
 163           [ok,more,explanations,worlds,help]: ok.
 164 \end{minted}
 165
 166 Using formula: $P(i_1|C_1)+P(i_2|C_2)(1-P(i_1|C_1))$ we can calculate the
 167 probability of the \textit{Alarm} variable using variable elimination. This
 168 results in the following answer:
 169 $$0.2*0.0027+0.95*0.0027*(1-0.2*0.0027)=0.00314699654673673$$
 170 TODOOOOOOOOOOO
 171
 172 \newpage
 173 \section{Burglary problem with extended information}
 174 $P(burglary)\cdot\left(
 175         P(\text{first house is holmes'})+
 176         P(\text{second house is holmes'})+
 177         P(\text{third house is holmes'})\right)=\\
 178 0.5102041\cdot\left(
 179         \frac{1}{10000}+
 180         \frac{9999}{10000}\cdot\frac{1}{9999}+
 181         \frac{9999}{10000}\cdot\frac{9998}{9999}\cdot\frac{1}{9998}\right)=
 182 \frac{3}{19600}\approx0.000153$
 183
 184 \section{Bayesian networks}
 185 A bayesian network representation of the burglary problem with a multitude of
 186 houses and burglars is possible but would be very big and tedious because all
 187 the constraints about the burglars must be incorporated in the network.
 188 The network would look something like in figere~\ref{bnnetworkhouses}
 189 accompanied with the probability distributions below.
 190
 191 \begin{tabular}{|l|l|}
 192         \hline
 193         Joe &\\
 194         \hline
 195         T & $\nicefrac{5}{7}$\\
 196         F & $\nicefrac{2}{7}$\\
 197         \hline
 198 \end{tabular}
 199 \begin{tabular}{|l|l|}
 200         \hline
 201         William &\\
 202         \hline
 203         T & $\nicefrac{5}{7}$\\
 204         F & $\nicefrac{2}{7}$\\
 205         \hline
 206 \end{tabular}
 207 \begin{tabular}{|l|l|}
 208         \hline
 209         Jack & \\
 210         \hline
 211         T & $\nicefrac{5}{7}$\\
 212         F & $\nicefrac{2}{7}$\\
 213         \hline
 214 \end{tabular}
 215 \begin{tabular}{|l|l|}
 216         \hline
 217         Averall & \\
 218         \hline
 219         T & $\nicefrac{5}{7}$\\
 220         F & $\nicefrac{2}{7}$\\
 221         \hline
 222 \end{tabular}
 223
 224 \begin{tabular}{|llll|ll|}
 225         \hline
 226         & & & & Burglary &\\
 227         Joe & William & Jack & Averall & T & F\\
 228         \hline
 229         F& F& F& F & $0$ & $1$\\
 230         F& F& F& T & $0$ & $1$\\
 231         F& F& T& F & $0$ & $1$\\
 232         F& F& T& T & $0$ & $1$\\
 233         F& T& F& F & $0$ & $1$\\
 234         F& T& F& T & $0$ & $1$\\
 235         F& T& T& F & $0$ & $1$\\
 236         F& T& T& T & $0$ & $1$\\
 237         T& F& F& F & $0$ & $1$\\
 238         T& F& F& T & $0$ & $1$\\
 239         T& F& T& F & $1$ & $0$\\
 240         T& F& T& T & $0$ & $1$\\
 241         T& T& F& F & $1$ & $0$\\
 242         T& T& F& T & $0$ & $1$\\
 243         T& T& T& F & $1$ & $0$\\
 244         T& T& T& T & $1$ & $0$\\
 245         \hline
 246 \end{tabular}
 247 \begin{tabular}{|lll|}
 248         \hline
 249         & Holmes &\\
 250         Burglary & T & F\\
 251         \hline
 252         T & $0.000153$ & $0.999847$\\
 253         F & $0$ & $1$\\
 254         \hline
 255 \end{tabular}
 256
 257 \begin{figure}[H]
 258         \caption{Bayesian network of burglars and houses}
 259         \label{bnnetworkhouses}
 260         \centering
 261         \includegraphics[scale=0.5]{d2.eps}
 262 \end{figure}