1 \chapter{Probabilistic representation and reasoning (and burglars)
}
2 \section{Formal description
}
3 In our representation of the model we introduced a
\textit{Noisy OR
} to
4 represent the causal independence of
\textit{Burglar
} and
\textit{Earthquake
}
5 on
\textit{Alarm
}. The representation of the network is displayed in
6 Figure~
\ref{bnetwork21
}
9 \caption{Bayesian network alarmsystem
}
12 \includegraphics[scale=
0.5]{d1.eps
}
15 Days were chosen as unit to model the story. Calculation for the probability of a
\textit{Burglar
} event happening at some day is then (assuming a gregorian
16 calendar and leap days):
17 $$
\frac{1}{365 +
0.25 -
0.01 -
0.0025}=
\frac{1}{365.2425}$$
19 The resultant probability distributions can be found in table ~
\ref{probdist
}, in order to avoid a unclear graph.
23 \begin{tabular
}{|l|l|
}
32 \begin{tabular
}{|l|l|
}
41 \begin{tabular
}{|l|ll|
}
43 &
\multicolumn{2}{c|
}{$I_1$
}\\
50 \begin{tabular
}{|l|ll|
}
52 &
\multicolumn{2}{c|
}{$I_2$
}\\
59 \begin{tabular
}{|ll|ll|
}
61 &&
\multicolumn{2}{c|
}{Alarm
}\\
62 $I_1$ & $I_2$ & T & F\\
71 \begin{tabular
}{|l|ll|
}
73 &
\multicolumn{2}{c|
}{Watson
}\\
80 \begin{tabular
}{|l|ll|
}
82 &
\multicolumn{2}{c|
}{Gibbons
}\\
89 \begin{tabular
}{|l|ll|
}
91 &
\multicolumn{2}{c|
}{Radio
}\\
94 T & $
0.9998$ & $
0.0002$\\
95 F & $
0.0002$ & $
0.9998$\\
100 \section{Implementation
}
101 This distribution results in the
\textit{AILog
} code in Listing~
\ref{alarm.ail
}
106 \inputminted[linenos,fontsize=
\footnotesize]{prolog
}{./src/alarm.ail
}
110 Using the following queries the probabilities or as follows:\\
111 \begin{enumerate
}[a)
]
112 \item $P(
\text{Burglary
})=
113 0.002737757092501968$
114 \item $P(
\text{Burglary
}|
\text{Watson called
})=
115 0.005321803679438259$
116 \item $P(
\text{Burglary
}|
\text{Watson called
}\wedge\text{Gibbons called
})=
118 \item $P(
\text{Burglary
}|
\text{Watson called
}\wedge\text{Gibbons called
}
119 \wedge\text{Radio
})=
0.01179672476662423$
123 \begin{minted
}[fontsize=
\footnotesize]{prolog
}
124 ailog: predict burglar.
125 Answer: P(burglar|Obs)=
0.002737757092501968.
126 [ok,more,explanations,worlds,help
]: ok.
128 ailog: observe watson.
129 Answer: P(watson|Obs)=
0.4012587986186947.
130 [ok,more,explanations,worlds,help
]: ok.
132 ailog: predict burglar.
133 Answer: P(burglar|Obs)=
[0.005321803679438259,
0.005321953115441623].
134 [ok,more,explanations,worlds,help
]: ok.
136 ailog: observe gibbons.
137 Answer: P(gibbons|Obs)=
[0.04596053565368094,
0.045962328885721306].
138 [ok,more,explanations,worlds,help
]: ok.
140 ailog: predict burglar.
141 Answer: P(burglar|Obs)=
[0.11180941544755249,
0.1118516494624678].
142 [ok,more,explanations,worlds,help
]: ok.
144 ailog: observe radio.
145 Answer: P(radio|Obs)=
[0.02582105837443645,
0.025915745316785182].
146 [ok,more,explanations,worlds,help
]: ok.
148 ailog: predict burglar.
149 Answer: P(burglar|Obs)=
[0.01179672476662423,
0.015584580594335082].
150 [ok,more,explanations,worlds,help
]: ok.
154 \section{Comparison with manual calculation
}
155 Querying the
\textit{Alarm
} variable gives the following answer
156 \begin{minted
}{prolog
}
157 ailog: predict alarm.
158 Answer: P(alarm|Obs)=
0.0031469965467367292.
160 [ok,more,explanations,worlds,help
]: ok.
163 Using formula: $P(i_1|C_1)+P(i_2|C_2)(
1-P(i_1|C_1))$ we can calculate the
164 probability of the
\textit{Alarm
} variable using variable elimination. This
165 results in the following answer:
166 $$
0.2*
0.0027+
0.95*
0.0027*(
1-
0.2*
0.0027)=
0.00314699654673673$$
170 \section{Burglary problem with extended information
}
171 $P(burglary)
\cdot\left(
172 P(
\text{first house is holmes'
})+
173 P(
\text{second house is holmes'
})+
174 P(
\text{third house is holmes'
})
\right)=\\
177 \frac{9999}{10000}\cdot\frac{1}{9999}+
178 \frac{9999}{10000}\cdot\frac{9998}{9999}\cdot\frac{1}{9998}\right)=
179 \frac{3}{19600}\approx0.000153$
181 \section{Bayesian networks
}
182 A bayesian network representation of the burglary problem with a multitude of
183 houses and burglars is possible but would be very big and tedious because all
184 the constraints about the burglars must be incorporated in the network.
185 The network would look something like in figere~
\ref{bnnetworkhouses
}
186 accompanied with the probability distributions below.
188 \begin{tabular
}{|l|l|
}
192 T & $
\nicefrac{5}{7}$\\
193 F & $
\nicefrac{2}{7}$\\
196 \begin{tabular
}{|l|l|
}
200 T & $
\nicefrac{5}{7}$\\
201 F & $
\nicefrac{2}{7}$\\
204 \begin{tabular
}{|l|l|
}
208 T & $
\nicefrac{5}{7}$\\
209 F & $
\nicefrac{2}{7}$\\
212 \begin{tabular
}{|l|l|
}
216 T & $
\nicefrac{5}{7}$\\
217 F & $
\nicefrac{2}{7}$\\
221 \begin{tabular
}{|llll|ll|
}
224 Joe & William & Jack & Averall & T & F\\
226 F& F& F& F & $
0$ & $
1$\\
227 F& F& F& T & $
0$ & $
1$\\
228 F& F& T& F & $
0$ & $
1$\\
229 F& F& T& T & $
0$ & $
1$\\
230 F& T& F& F & $
0$ & $
1$\\
231 F& T& F& T & $
0$ & $
1$\\
232 F& T& T& F & $
0$ & $
1$\\
233 F& T& T& T & $
0$ & $
1$\\
234 T& F& F& F & $
0$ & $
1$\\
235 T& F& F& T & $
0$ & $
1$\\
236 T& F& T& F & $
1$ & $
0$\\
237 T& F& T& T & $
0$ & $
1$\\
238 T& T& F& F & $
1$ & $
0$\\
239 T& T& F& T & $
0$ & $
1$\\
240 T& T& T& F & $
1$ & $
0$\\
241 T& T& T& T & $
1$ & $
0$\\
244 \begin{tabular
}{|lll|
}
249 T & $
0.000153$ & $
0.999847$\\
255 \caption{Bayesian network of burglars and houses
}
256 \label{bnnetworkhouses
}
258 %\includegraphics[scale=0.5]{d2.eps}