\documentclass[pdf,bndg,slideColor,colorBG]{prosper}

\input{preamble}

\title{Bayesian Networks and Decision Graphs}
\subtitle{}
\author{}
\email{}
\institution{Chapter 9}
\slideCaption{Chapter 9}


\begin{document}
\maketitle






%####################################################################################
%# A small quiz
%####################################################################################

\PDForPS{
\overlays{3}{
\begin{slide}{A small quiz}

\fromSlide*{1}{
Which of the following two lotteries would you prefer?:
\begin{itemize}
\item Lottery $A = [\$1 \rm{mill.}]$,
\item Lottery $B = 0.1[\$5 \rm{mill.}] + 0.89[\$1 \rm{mill.}] + 0.01[\$0]$.
\end{itemize}
\ \\
\ \\

}
\fromSlide*{2}{
What about these two?:

\begin{itemize}
\item Lottery $C = 0.11[\$1 \rm{mill.}] + 0.89[\$0]$,
\item  Lottery $D = 0.1[\$5 \rm{mill.}] + 0.9[\$0]$.
\end{itemize}
\ \\


}
\fromSlide*{3}{
\centerline{
\medred{Is this the rational choice?}
}

}
\end{slide}}
}
{
\begin{slide}{A small quiz}

Which of the following two lotteries would you prefer?:
\begin{itemize}
\item Lottery $A = [\$1 \rm{mill.}]$,
\item Lottery $B = 0.1[\$5 \rm{mill.}] + 0.89[\$1 \rm{mill.}] + 0.01[\$0]$.
\end{itemize}

What about these two?:

\begin{itemize}
\item Lottery $C = 0.11[\$1 \rm{mill.}] + 0.89[\$0]$,
\item  Lottery $D = 0.1[\$5 \rm{mill.}] + 0.9[\$0]$.
\end{itemize}

Is this the rational choice?

\end{slide}


}



%####################################################################################
%# Reverse the directions?
%####################################################################################

\begin{slide}{Reverse the directions?}

\ul{Consider the following model:}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(0,0)(14,3)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{sl}{Sleepy}

\arc{fl}{fe}
\arc{fe}{sl}
%\showgrid
\end{pspicture}
\end{center}
\ \\
The probability distributions $P(\VarM{Sleepy})$,
$P(\VarM{Fever}|\VarM{Sleepy})$ and $P(\VarM{Flue}|\VarM{Fever})$ can
be calculated from the model above and used in the model below.

\ \\
\ \\

\begin{center}
\begin{pspicture}(0,0)(14,3)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{sl}{Sleepy}

\arc{fe}{fl}
\arc{sl}{fe}
%\showgrid
\end{pspicture}
\end{center}

\centerline{
\Red{So is there any difference??}
}
\end{slide}


%####################################################################################
%# Decisions
%####################################################################################

\begin{slide}{Decisions}

Taking the temperature and setting the temperature can be seen as a
test decision and an action decision, respectively.

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(0,0)(14,7)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{sl}{Sleepy}

\arc{fl}{fe}
\arc{fe}{sl}

\Fbox[0.6cm]{10}{5}{a}{Action}
\Ftria[0.6cm]{4}{5}{t}{Test}

\arc{fe}{t}
\arc{a}{fe}
%\showgrid
\end{pspicture}
\end{center}

\ul{Impacts from the decisions:}
\begin{itemize}
\item $ $ \Red{Tests}: Both directions
\item $ $ \Red{Actions}: \ul{With} the direction only. 
\end{itemize}


\end{slide}

%####################################################################################
%# Poker again
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Poker again}
\fromSlide*{1}{
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

Consider the poker example again:

\begin{center}
\begin{pspicture}(1,1)(25,10)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}$}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}

\rput[l](19,4){Why request this?}

\psline[linecolor=magenta,linestyle=dashed]{->}(22,5)(19,7)


%\showgrid
\end{pspicture}
\end{center}
}
\fromSlide*{2}{
\ul{\Red{Fold or call?}}
\begin{itemize}
\item Both placed $1\$$
\item She has placed $1\$$ more
\item $ $ \Red{fold} $\Rightarrow$ she takes the pot
\item $ $ \Red{call} $\Rightarrow$ place $1\$$ $\Rightarrow$ best hand takes the pot
\end{itemize}
}
\end{slide}
}
}
{
\begin{slide}{Poker again}

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(25,10)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}$}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}

\rput[l](19,4){Why request this?}

\psline[linecolor=magenta,linestyle=dashed]{->}(22,5)(19,7)


%\showgrid
\end{pspicture}
\end{center}

\ul{\Red{Fold or call?}}
\begin{itemize}
\item Both placed $1\$$
\item She has placed $1\$$ more
\item $ $ \Red{fold} $\Rightarrow$ she takes the pot
\item $ $ \Red{call} $\Rightarrow$ place $1\$$ $\Rightarrow$ best hand takes the pot
\end{itemize}

\end{slide}

}



%####################################################################################
%# Call or fold?
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Call or fold?}
\fromSlide*{1}{
This decision problem can be represented graphically by extending the
BN with a \Red{decision node} and a \Purple{utility node}:
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(30,10)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}$}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}

\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\rput(11,9){\rnode{e1}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

\rput(2,2){\rnode{e2}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

\rput(8,2){\rnode{e3}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}

\rput[l](21.5,3){
\begin{tabular}{cc|cc}
 &  & \multicolumn{2}{c}{$\mbox{\Red{D}}$}\\ 
 & & fold & call  \\ \hline
\multirow{3}{0.1cm}{$\VarM{BH}$} & I & $-1$ & $2$ \\
 & op.\ & $-1$ & $-2$ \\
 & draw\ & $-1$ & $0$ \\
\multicolumn{4}{c}{$\Purple{\mbox{U}}(\VarM{BH},\Red{\mbox{D}})$}
\end{tabular}
}

%\showgrid
\end{pspicture}
\end{center}
}
\fromSlide*{2}{
\ \\
\ \\
\ \\

The expected utility of call:
\[
\begin{split}
\mbox{EU}(\mbox{\Red{call}}|\Red{e})&=2\cdot P(\VarM{BH}=\mbox{I}|\Red{e})-2\cdot
P(\VarM{BH}=\mbox{op.}|\Red{e})+0\cdot
P(\VarM{BH}=\mbox{draw}|\Red{e})\\
&=\sum _{\mbox{BH}}\Purple{\mbox{U}}(\VarM{BH},\Red{\mbox{call}})P(\VarM{BH}|\Red{e})
\end{split}
\]
}
\end{slide}
}
}
{
\begin{slide}{Call or fold?}

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(30,10)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}$}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}

\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\rput(11,9){\rnode{e1}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

\rput(2,2){\rnode{e2}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

\rput(8,2){\rnode{e3}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}

\rput[l](21.5,3){
\begin{tabular}{cc|cc}
 &  & \multicolumn{2}{c}{$\mbox{\Red{D}}$}\\ 
 & & fold & call  \\ \hline
\multirow{3}{0.1cm}{$\VarM{BH}$} & I & $-1$ & $2$ \\
 & op.\ & $-1$ & $-2$ \\
 & draw\ & $-1$ & $0$ \\
\multicolumn{4}{c}{$\Purple{\mbox{U}}(\VarM{BH},\Red{\mbox{D}})$}
\end{tabular}
}

%\showgrid
\end{pspicture}
\end{center}

\ \\
\ \\
\ \\

The expected utility of call:
\[
\begin{split}
\mbox{EU}(\mbox{\Red{call}}|\Red{e})&=2\cdot P(\VarM{BH}=\mbox{I}|\Red{e})-2\cdot
P(\VarM{BH}=\mbox{op.}|\Red{e})+0\cdot
P(\VarM{BH}=\mbox{draw}|\Red{e})\\
&=\sum _{\mbox{BH}}\Purple{\mbox{U}}(\VarM{BH},\Red{\mbox{call}})P(\VarM{BH}|\Red{e})
\end{split}
\]

\end{slide}
}

%####################################################################################
%# Mildew
%####################################################################################

\PDForPS{
\overlays{3}{
\begin{slide}{Mildew}
\fromSlide*{1}{
Two months before the harvest the farmer observes the state, \Blue{Q},
of his wheat field, and he can check whether the field is attacked by
mildew, \Blue{M}. If there is a mildew attack he can decide for a
treatment with fungicides.

}
\fromSlide*{2}{
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(0,1)(14,13)

\Fnode{2}{2}{om}{OM}
\Fnode{2}{5}{m}{M}
\Fnode{2}{8}{oq}{OQ}
\Fnode{2}{11}{q}{Q}

\Fovalnode[0.7cm]{7}{11}{h}{Harvest}
\Fnode{7}{5}{mm}{$\mbox{M}^*$}

\Fbox{7}{2}{a}{A}

\Fdiam{12}{2}{c}{C}
\Fdiam{12}{11}{u}{U}

\rput(0,2){\rnode{e1}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{om}

\rput(0,8){\rnode{e2}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{oq}

\arc{q}{oq}
\arc{q}{h}
\arc{h}{u}
\arc{mm}{h}
\arc{m}{mm}
\arc{m}{om}
\arc{a}{mm}
\arc{a}{c}

%\showgrid
\end{pspicture}
\end{center}


}
\fromSlide*{3}{
\[
EU(\Red{A}|\Red{e})=\Purple{\mbox{C}}(\DecM{A})+\sum _{\mbox{\Blue{Harvest}}}\UM{U}(\VarM{Harvest})P(\VarM{Harvest}|\DecM{A},\Red{e})
\]


}


\end{slide}
}
}
{
\begin{slide}{Mildew}
Two months before the harvest the farmer observes the state, \Blue{Q},
of his wheat field, and he can check whether the field is attacked by
mildew, \Blue{M}. If there is a mildew attack he can decide for a
treatment with fungicides.
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(0,1)(14,13)

\Fnode{2}{2}{om}{OM}
\Fnode{2}{5}{m}{M}
\Fnode{2}{8}{oq}{OQ}
\Fnode{2}{11}{q}{Q}

\Fovalnode[0.7cm]{7}{11}{h}{Harvest}
\Fnode{7}{5}{mm}{$\mbox{M}^*$}

\Fbox{7}{2}{a}{A}

\Fdiam{12}{2}{c}{C}
\Fdiam{12}{11}{u}{U}

\rput(0,2){\rnode{e1}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{om}

\rput(0,8){\rnode{e2}{}}
\ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{oq}

\arc{q}{oq}
\arc{q}{h}
\arc{h}{u}
\arc{mm}{h}
\arc{m}{mm}
\arc{m}{om}
\arc{a}{mm}
\arc{a}{c}

%\showgrid
\end{pspicture}
\end{center}

\[
EU(\Red{A}|\Red{e})=\Purple{\mbox{C}}(\DecM{A})+\sum _{\mbox{\Blue{Harvest}}}\UM{U}(\VarM{Harvest})P(\VarM{Harvest}|\DecM{A},\Red{e})
\]

\end{slide}

}

%####################################################################################
%# One action in general
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{One action in general}
\fromSlide*{1}{
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,-1)(30,10)

\Fbox{3}{6}{d}{$D$}
\Fdiam{3}{2}{u1}{$U_l$}
\Fnode{7}{7}{x1}{}
\Fnode{7}{3}{x2}{}
\Fnode{12}{7}{x3}{}
\Fnode{12}{3}{x4}{}
\Fnode{18}{9}{x5}{}
\Fnode{18}{5.5}{x6}{}
\Fnode{18}{2}{x7}{}
\Fnode{18}{-1.5}{x8}{}

\Fdiam{22}{7.25}{u2}{$U_j$}
\Fdiam{22}{0.25}{u3}{$U_k$}

\arc{x5}{u2}
\arc{x6}{u2}
\arc{x7}{u3}
\arc{x8}{u3}
\arc{d}{u1}
\arc{d}{x1}
\arc{d}{x2}
\arc{x2}{u1}

\rput(10,9){\rnode{y1}{}}
\rput(14,9){\rnode{y2}{}}
\rput(10,5){\rnode{y3}{}}
\rput(10,1){\rnode{y4}{}}

\arc{y1}{x3}
\arc{y2}{x3}
\arc{x3}{y3}

\ncline[linestyle=dashed,linecolor=red]{->}{y4}{x4}

\ncline[linestyle=dotted,linecolor=red]{-}{x5}{x6}
\ncline[linestyle=dotted,linecolor=red]{-}{x6}{x7}
\ncline[linestyle=dotted,linecolor=red]{-}{x7}{x8}

\rput[l](23,4){$\Purple{U_i}$ has domain $\Blue{\mathcal X_i}$}


%\showgrid
\end{pspicture}
\end{center}
}
\fromSlide*{2}{


\vspace*{-0.1cm}
\[
\begin{split}
EU(\Red{D}|\Red{e})&=\sum
_{\Blue{\mathcal X_1}}\Purple{U_1}(\Blue{\mathcal X_1})P(\Blue{\mathcal X_1}|\Red{D},\Red{e})+\cdots +\sum
_{\Blue{\mathcal X_n}}\Purple{U_n}(\Blue{\mathcal X_n})P(\Blue{\mathcal X_n}|\Red{D},\Red{e}) 
\end{split}
\]
\vspace*{-0.2cm}
Choose an action with largest EU:
\[
Opt(\Red{D}|\Red{e})=\arg \max_{\Red{D}}EU(\Red{D}|\Red{e})
\]



}
\end{slide}
}
}
{
\begin{slide}{One action in general}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,-1)(30,10)

\Fbox{3}{6}{d}{$D$}
\Fdiam{3}{2}{u1}{$U_l$}
\Fnode{7}{7}{x1}{}
\Fnode{7}{3}{x2}{}
\Fnode{12}{7}{x3}{}
\Fnode{12}{3}{x4}{}
\Fnode{18}{9}{x5}{}
\Fnode{18}{5.5}{x6}{}
\Fnode{18}{2}{x7}{}
\Fnode{18}{-1.5}{x8}{}

\Fdiam{22}{7.25}{u2}{$U_j$}
\Fdiam{22}{0.25}{u3}{$U_k$}

\arc{x5}{u2}
\arc{x6}{u2}
\arc{x7}{u3}
\arc{x8}{u3}
\arc{d}{u1}
\arc{d}{x1}
\arc{d}{x2}
\arc{x2}{u1}

\rput(10,9){\rnode{y1}{}}
\rput(14,9){\rnode{y2}{}}
\rput(10,5){\rnode{y3}{}}
\rput(10,1){\rnode{y4}{}}

\arc{y1}{x3}
\arc{y2}{x3}
\arc{x3}{y3}

\ncline[linestyle=dashed,linecolor=red]{->}{y4}{x4}

\ncline[linestyle=dotted,linecolor=red]{-}{x5}{x6}
\ncline[linestyle=dotted,linecolor=red]{-}{x6}{x7}
\ncline[linestyle=dotted,linecolor=red]{-}{x7}{x8}

\rput[l](23,4){$\Purple{U_i}$ has domain $\mathcal X_i$}


%\showgrid
\end{pspicture}
\end{center}

\vspace*{-0.1cm}
\[
\begin{split}
EU(\Red{D}|\Red{e})&=\sum
_{\mathcal X_1}\Purple{U_1}(\mathcal X_1)P(\mathcal X_1|\Red{D},\Red{e})+\cdots +\sum
_{\mathcal X_n}\Purple{U_n}(\mathcal X_n)P(\mathcal X_n|\Red{D},\Red{e}) 
\end{split}
\]
\vspace*{-0.2cm}
Choose an action with largest EU:
\[
Opt(\Red{D}|\Red{e})=\arg \max_{\Red{D}}EU(\Red{D}|\Red{e})
\]

\end{slide}
}

%####################################################################################
%# Utilities without money 
%####################################################################################

\begin{slide}{Utilities without money}

\ul{\Green{Two courses:}} Graph algorithms (\Blue{GA}) and \Blue{DSS}\newline
\newline
\ul{\Green{Marks:}} $0,1,2,3,4,5$ ($\geq 2$ is a pass)\newline\newline
\ul{\Green{Effort:}} Keep pace (kp), slow down (sd), follow
superficially (fs)
\ \\
\ \\
\ \\

\begin{center}
\begin{tabular}{cc}
\begin{tabular}{cc|ccc}
 & & \multicolumn{3}{c}{\Blue{Effort}}\\
 & & kp & sd & fs \\ \hline
\multirow{5}{0.2cm}{\Blue{GA}} 
& $0$ & $0$ & $0$ & $0.1$ \\
& $1$ & $0.1$ & $0.2$ & $0.1$ \\
& $2$ & $0.1$ & $0.1$ & $0.4$ \\
& $3$ & $0.2$ & $0.4$ & $0.2$ \\
& $4$ & $0.4$ & $0.2$ & $0.2$ \\
& $5$ & $0.2$ & $0.1$ & $0$ \\
\multicolumn{5}{c}{$P(\VarM{GA}|\VarM{Effort})$}
\end{tabular} 
&
\begin{tabular}{cc|ccc}
 & & \multicolumn{3}{c}{\Blue{Effort}}\\
 & & kp & sd & fs \\ \hline
\multirow{5}{0.2cm}{\Blue{DSS}} 
& $0$ & $0$ & $0$ & $0.1$ \\
& $1$ & $0$ & $0.1$ & $0.2$ \\
& $2$ & $0.1$ & $0.2$ & $0.2$ \\
& $3$ & $0.2$ & $0.2$ & $0.3$ \\
& $4$ & $0.4$ & $0.4$ & $0.2$ \\
& $5$ & $0.3$ & $0.1$ & $0$ \\
\multicolumn{5}{c}{$P(\VarM{DSS}|\VarM{Effort})$}
\end{tabular} 
\end{tabular}
\end{center}
\ \\
\ \\
\ \\


\centerline{
Max-score?\hspace*{1cm}Max-pass?\hspace*{1cm}Otherwise?
}


\end{slide}


%####################################################################################
%# Marks as utilities? 
%####################################################################################

\PDForPS{
\overlays{3}{
\begin{slide}{Marks as utilities?}
\fromSlide*{1}{
\begin{center}
\begin{tabular}{cc}
\begin{tabular}{cc|ccc}
 & & \multicolumn{3}{c}{\Blue{Effort}}\\
 & & kp & sd & fs \\ \hline
\multirow{5}{0.2cm}{\Blue{GA}} 
& $0$ & $0$ & $0$ & $0.1$ \\
& $1$ & $0.1$ & $0.2$ & $0.1$ \\
& $2$ & $0.1$ & $0.1$ & $0.4$ \\
& $3$ & $0.2$ & $0.4$ & $0.2$ \\
& $4$ & $0.4$ & $0.2$ & $0.2$ \\
& $5$ & $0.2$ & $0.1$ & $0$ \\
\multicolumn{5}{c}{$P(\VarM{GA}|\VarM{Effort})$}
\end{tabular} 
&
\begin{tabular}{cc|ccc}
 & & \multicolumn{3}{c}{\Blue{Effort}}\\
 & & kp & sd & fs \\ \hline
\multirow{5}{0.2cm}{\Blue{DSS}} 
& $0$ & $0$ & $0$ & $0.1$ \\
& $1$ & $0$ & $0.1$ & $0.2$ \\
& $2$ & $0.1$ & $0.2$ & $0.2$ \\
& $3$ & $0.2$ & $0.2$ & $0.3$ \\
& $4$ & $0.4$ & $0.4$ & $0.2$ \\
& $5$ & $0.3$ & $0.1$ & $0$ \\
\multicolumn{5}{c}{$P(\VarM{DSS}|\VarM{Effort})$}
\end{tabular} 
\end{tabular}
\end{center}
\ \\


}
\[
\begin{split}
\fromSlide*{1}{
EU&(\mbox{kp,fs}) = \sum _{\mbox{m}\in
  \VarM{GA}}P(\mbox{m}|\mbox{kp})\mbox{m}+\sum _{\mbox{m}\in
  \VarM{DSS}}P(\mbox{m}|\mbox{fs})\mbox{m} \\}
&= (0.1\cdot 1+0.1\cdot 2 + 0.2\cdot 3 + 0.4\cdot 4+0.2\cdot 5) +
  (0.2\cdot 1+0.2\cdot 2 + 0.3\cdot 3 + 0.2\cdot 4) = 5.8\\
\fromSlide*{2}{EU&(\mbox{sd,sd}) = 6.1\\}
\fromSlide*{3}{EU&(\mbox{fs,kp}) = \ul{6.2} }
\end{split}
\]

\fromSlide*{3}{
\centerline{
\Red{
However, does the marks really reflect your utilities?
}}}

\end{slide}}}
{
\begin{slide}{Marks as utilities?}

\begin{center}
\begin{tabular}{cc}
\begin{tabular}{cc|ccc}
 & & \multicolumn{3}{c}{\Blue{Effort}}\\
 & & kp & sd & fs \\ \hline
\multirow{5}{0.2cm}{\Blue{GA}} 
& $0$ & $0$ & $0$ & $0.1$ \\
& $1$ & $0.1$ & $0.2$ & $0.1$ \\
& $2$ & $0.1$ & $0.1$ & $0.4$ \\
& $3$ & $0.2$ & $0.4$ & $0.2$ \\
& $4$ & $0.4$ & $0.2$ & $0.2$ \\
& $5$ & $0.2$ & $0.1$ & $0$ \\
\multicolumn{5}{c}{$P(\VarM{GA}|\VarM{Effort})$}
\end{tabular} 
&
\begin{tabular}{cc|ccc}
 & & \multicolumn{3}{c}{\Blue{Effort}}\\
 & & kp & sd & fs \\ \hline
\multirow{5}{0.2cm}{\Blue{DSS}} 
& $0$ & $0$ & $0$ & $0.1$ \\
& $1$ & $0$ & $0.1$ & $0.2$ \\
& $2$ & $0.1$ & $0.2$ & $0.2$ \\
& $3$ & $0.2$ & $0.2$ & $0.3$ \\
& $4$ & $0.4$ & $0.4$ & $0.2$ \\
& $5$ & $0.3$ & $0.1$ & $0$ \\
\multicolumn{5}{c}{$P(\VarM{DSS}|\VarM{Effort})$}
\end{tabular} 
\end{tabular}
\end{center}
\ \\


\[
\begin{split}
EU&(\mbox{kp,fs}) = \sum _{\mbox{m}\in
  \VarM{GA}}P(\mbox{m}|\mbox{kp})\mbox{m}+\sum _{\mbox{m}\in
  \VarM{DSS}}P(\mbox{m}|\mbox{fs})\mbox{m} \\
&= (0.1\cdot 1+0.1\cdot 2 + 0.2\cdot 3 + 0.4\cdot 4+0.2\cdot 5) +
  (0.2\cdot 1+0.2\cdot 2 + 0.3\cdot 3 + 0.2\cdot 4) = 5.8\\
EU&(\mbox{sd,sd}) = 6.1\\
EU&(\mbox{fs,kp}) = \ul{6.2} 
\end{split}
\]

\centerline{
\Red{
However, does the marks really reflect your utilities?
}}

\end{slide}
}


%####################################################################################
%# Subjective lotteries
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Subjective lotteries}
\fromSlide*{1}{
I consider $2$ as the worst mark (\Purple{utility $0$}) and $5$ as the best
mark (\Purple{utility $1$}). Now imagine the following lottery:
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(3,1)(16,8)

\rput(4,5){\rnode{a}{}}
\Fnode{8}{7}{2}{$2$}
\Fnode{8}{3}{5}{$5$}
\Fnode{14}{5}{x}{$X$}

\rput[c](5,4){$p$}
\rput[c](5,6.2){$1-p$}
\rput[c](15,6){$4$}


\arc{a}{2}
\arc{a}{5}
%\showgrid
\end{pspicture}
\end{center}
\vspace*{-0.3cm}
\centerline{
For which $p$ am I indifferent??
}
}
\fromSlide*{2}{



\begin{center}
The utility table: 
\begin{tabular}{|cccccc|}\hline
$0$ & $1$ & $2$  & $3$ & $4$ & $5$   \\  
$0.05$ & $0.1$ & $0$  & $0.6$ & $0.8$ & $1$   \\  \hline
\end{tabular}
\end{center}
\ \\
\ \\

\begin{center}
\begin{pspicture}(-5,1)(20,6)

\Fbox[0.5cm]{3}{3}{ef}{Effort}
\Fnode{6}{5}{ga}{GA}
\Fnode{6}{1}{dss}{DSS}
\Fdiam[0.6cm]{11}{5}{ug}{$U_{\mbox{GA}}$}
\Fdiam[0.6cm]{11}{1}{ud}{$U_{\mbox{DSS}}$}

\arc{ef}{ga}
\arc{ef}{dss}
\arc{ga}{ug}
\arc{dss}{ud}

\rput[r](2,1.5){(kp,fs),(sd,sd),(fs,kp)}

\rput[l](12,3){$EU(\DecM{Effort})=(1.015,\ul{1.07},1.035)$}
%\showgrid
\end{pspicture}
\end{center}


}
\end{slide}
}
}
{
\begin{slide}{Subjective lotteries}

I consider $2$ as the worst mark (utility $0$) and $5$ as the best
mark (utility $1$). Now imagine the following lottery:
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(3,1)(16,8)

\rput(4,5){\rnode{a}{}}
\Fnode{8}{7}{2}{$2$}
\Fnode{8}{3}{5}{$5$}
\Fnode{14}{5}{x}{$X$}

\rput[c](5,4){$p$}
\rput[c](5,6.2){$1-p$}
\rput[c](15,6){$4$}


\arc{a}{2}
\arc{a}{5}
%\showgrid
\end{pspicture}
\end{center}
\vspace*{-0.3cm}
\centerline{
For which $p$ am I indifferent??
}

\begin{center}
The utility table: 
\begin{tabular}{|cccccc|}\hline
$0$ & $1$ & $2$  & $3$ & $4$ & $5$   \\  
$0.05$ & $0.1$ & $0$  & $0.6$ & $0.8$ & $1$   \\  \hline
\end{tabular}
\end{center}
\ \\
\ \\

\begin{center}
\begin{pspicture}(-5,1)(20,6)

\Fbox[0.5cm]{3}{3}{ef}{Effort}
\Fnode{6}{5}{ga}{GA}
\Fnode{6}{1}{dss}{DSS}
\Fdiam[0.6cm]{11}{5}{ug}{$U_{\mbox{GA}}$}
\Fdiam[0.6cm]{11}{1}{ud}{$U_{\mbox{DSS}}$}

\arc{ef}{ga}
\arc{ef}{dss}
\arc{ga}{ug}
\arc{dss}{ud}

\rput[r](2,1.5){(kp,fs),(sd,sd),(fs,kp)}

\rput[l](12,3){$EU(\DecM{Effort})=(1.015,\ul{1.07},1.035)$}
%\showgrid
\end{pspicture}
\end{center}

\end{slide}
}

%####################################################################################
%# Instrumental rationality
%####################################################################################

\PDForPS{
\overlays{7}{
\begin{slide}{Instrumental rationality}

\begin{description}
\fromSlide*{1}{
\item[{\em 1. Reflexivity.}] For any lottery $A$, $A \succeq A$
}

\fromSlide*{2}{
\item [{\em 2. Completeness.}] For any pair $(A, B)$ of lotteries, $A \succeq
B$ or $B \succeq A$.
}

\fromSlide*{3}{
\item [{\em 3. Transitivity.}] If $A \succeq B$ and $B \succeq C$, then $A
\succeq C$.
}

\fromSlide*{4}{
\item [{\em 4. Preference increasing with probability.}] If $A \succeq B$ then
$\alpha A + (1-\alpha )B \succeq \beta A + (1-\beta )B$ if and only if
$\alpha \geq \beta$.
}

\fromSlide*{5}{
\item [{\em 5. Continuity.}] If $A \succeq B \succeq C$ then there exists
$\alpha \in [0, 1]$ such that $B \sim \alpha A + (1-\alpha )C$
}

\fromSlide*{6}{
\item [{\em 6. Independence.}] If $C = \alpha A + (1-\alpha )B$ and $A \sim
D$, then $C \sim (\alpha D + (1-\alpha )B)$.
}
\end{description}
\ \\
\ \\

\fromSlide*{7}{
{\bf Theorem:} For an individual who acts according to a preference ordering satisfying rules 1-6
above, there exists a utility function over the outcomes s.t.\ the
expected utility is maximized. 
}
\end{slide}
}}
{
\begin{slide}{Instrumental rationality}

\begin{description}
\item[{\em 1. Reflexivity.}] For any lottery $A$, $A \succeq A$


\item [{\em 2. Completeness.}] For any pair $(A, B)$ of lotteries, $A \succeq
B$ or $B \succeq A$.


\item [{\em 3. Transitivity.}] If $A \succeq B$ and $B \succeq C$, then $A
\succeq C$.


\item [{\em 4. Preference increasing with probability.}] If $A \succeq B$ then
$\alpha A + (1-\alpha )B \succeq \beta A + (1-\beta )B$ if and only if
$\alpha \geq \beta$.


\item [{\em 5. Continuity.}] If $A \succeq B \succeq C$ then there exists
$\alpha \in [0, 1]$ such that $B \sim \alpha A + (1-\alpha )C$


\item [{\em 6. Independence.}] If $C = \alpha A + (1-\alpha )B$ and $A \sim
D$, then $C \sim (\alpha D + (1-\alpha )B)$.

\end{description}


{\bf Theorem:} For an individual who acts according to a preference ordering satisfying rules 1-6
above, there exists a utility function over the outcomes s.t.\ the
expected utility is maximized. 

\end{slide}

}


%####################################################################################
%# The quiz
%####################################################################################

\PDForPS{
\overlays{3}{
\begin{slide}{Are you rational}
\fromSlide*{1}{
Recall:
\begin{itemize}
\item Lottery $A = [\$1 \rm{mill.}]$,
\item Lottery $B = 0.1[\$5 \rm{mill.}] + 0.89[\$1 \rm{mill.}] + 0.01[\$0]$.
\item Lottery $C = 0.11[\$1 \rm{mill.}] + 0.89[\$0]$,
\item  Lottery $D = 0.1[\$5 \rm{mill.}] + 0.9[\$0]$.
\end{itemize}
\ \\
\ \\


}
\fromSlide*{2}{
Let $U(5 mill) = 1, U(0) = 0, U(1 mill) = u$. If
you prefer $A$ over $B$ we get
\[
 u > 0.1 + 0.89u \qquad \Leftrightarrow \qquad u>\frac{10}{11}.
\]

}
\fromSlide*{3}{
Hence,

\[
EU(C) = 0.11u > 0.11\frac{10}{11} = 0.1 = EU(D),
\]
and $C$ should therefore be preferred over $D$.


}

\end{slide}}}
{

\begin{slide}{Are you rational}
Recall:
\begin{itemize}
\item Lottery $A = [\$1 \rm{mill.}]$,
\item Lottery $B = 0.1[\$5 \rm{mill.}] + 0.89[\$1 \rm{mill.}] + 0.01[\$0]$.
\item Lottery $C = 0.11[\$1 \rm{mill.}] + 0.89[\$0]$,
\item  Lottery $D = 0.1[\$5 \rm{mill.}] + 0.9[\$0]$.
\end{itemize}

Let $U(5 mill) = 1, U(0) = 0, U(1 mill) = u$. If
you prefer $A$ from $B$ we get
\[
 u > 0.1 + 0.89u \qquad \Leftrightarrow \qquad u>\frac{10}{11}.
\]
Hence,

\[
EU(C) = 0.11u > 0.11\frac{10}{11} = 0.1 = EU(D),
\]
and $C$ should therefore be preferred over $D$.


\end{slide}
}



%####################################################################################
%# Decision trees
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Decision trees}
\fromSlide*{1}{
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,4)

\Fnode{2}{2}{i}{Inf}
\Fnode{7}{2}{t}{Test}
\arc{i}{t}
\pspolygon[linecolor=black](0.5,0.5)(8.5,0.5)(8.5,3.5)(0.5,3.5)

%\showgrid
\end{pspicture}

\vspace*{-2.3cm}
\psset{treesep=0.25cm}
\psset{levelsep=2cm}
\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect{T}}
{
	\pstree{\Foval[0.5cm]{Test}^{y}}
	{
		\pstree{\Frect[0.6cm]{Pour?}^{pos}}
		{
		  
		  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
		  {
		    \Fdia[0.4cm]{$99.94$}^{n}
		    \Fdia[0.4cm]{$-0.06$}_{y}
		  }
		  \Fdia[0.4cm]{$97.94$}_{throw}
		  
		}
		\pstree{\Frect[0.6cm]{Pour?}_{neg}}
		       {
			 \pstree{\Foval[0.5cm]{Inf?}^{pour}}
				{
				  \Fdia[0.4cm]{$99.94$}^{n}
				  \Fdia[0.4cm]{$-0.06$}_{y}
				}
				\Fdia[0.4cm]{$97.94$}_{throw}
		       }
	}
	\pstree{\Frect[0.6cm]{Pour?}_{n}}
	{
	  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
	  {
	    \Fdia[0.4cm]{$0$}^{y}
	    \Fdia[0.4cm]{$100$}_{n}
	  }
	  \Fdia[0.4cm]{$98$}_{throw}
	}

}
}
\end{center}
}
\fromSlide*{2}{
\ \\
\ \\

Branches from chance nodes, $\Var{\bigcirc}$, shall be labeled with the
probability of the branch given the path down to the node. The
probabilities can be found from the model \fbox{$\Var{\bigcirc}\Green{ \rightarrow} \Var{\bigcirc}$}.

}
\end{slide}}
}
{
\begin{slide}{Decision trees}

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,4)

\Fnode{2}{2}{i}{Inf}
\Fnode{7}{2}{t}{Test}
\arc{i}{t}
\pspolygon[linecolor=black](0.5,0.5)(8.5,0.5)(8.5,3.5)(0.5,3.5)

%\showgrid
\end{pspicture}

\vspace*{-2.3cm}
\psset{treesep=0.25cm}
\psset{levelsep=2cm}
\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect{T}}
{
	\pstree{\Foval[0.5cm]{Test}^{y}}
	{
		\pstree{\Frect[0.6cm]{Pour?}^{pos}}
		{
		  
		  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
		  {
		    \Fdia[0.4cm]{$99.94$}^{n}
		    \Fdia[0.4cm]{$-0.06$}_{y}
		  }
		  \Fdia[0.4cm]{$97.94$}_{throw}
		  
		}
		\pstree{\Frect[0.6cm]{Pour?}_{neg}}
		       {
			 \pstree{\Foval[0.5cm]{Inf?}^{pour}}
				{
				  \Fdia[0.4cm]{$99.94$}^{n}
				  \Fdia[0.4cm]{$-0.06$}_{y}
				}
				\Fdia[0.4cm]{$97.94$}_{throw}
		       }
	}
	\pstree{\Frect[0.6cm]{Pour?}_{n}}
	{
	  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
	  {
	    \Fdia[0.4cm]{$0$}^{y}
	    \Fdia[0.4cm]{$100$}_{n}
	  }
	  \Fdia[0.4cm]{$98$}_{throw}
	}

}
}
\end{center}

Branches from chance nodes, $\Var{\bigcirc}$, shall be labeled with the
probability of the branch given the path down to the node. The
probabilities can be found from the model \fbox{$\Var{\bigcirc}\Green{ \rightarrow} \Var{\bigcirc}$}.


\end{slide}

}

%####################################################################################
%# Solving decision trees I
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Solving decision trees I}
\fromSlide*{1}{
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,4)

\Fnode{2}{2}{i}{Inf}
\Fnode{7}{2}{t}{Test}
\arc{i}{t}
\pspolygon[linecolor=black](0.5,0.5)(8.5,0.5)(8.5,3.5)(0.5,3.5)

%\showgrid
\end{pspicture}

\vspace*{-2.3cm}
\psset{treesep=0.25cm}
\psset{levelsep=2.5cm}
\psset{npos=0.5}
\psset{labelsep=0.5pt}
\psset{nrot=:U}

\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect{T}}
{
	\pstree{\Foval[0.5cm]{Test}^{y}}
	{
		\pstree{\Frect[0.6cm]{Pour?}\naput{pos: $0.0107$}}
		{
		  
		  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
		  {
		    \Fdia[0.4cm]{$99.94$}\naput{n: $0.9351$}
		    \Fdia[0.4cm]{$-0.06$}\nbput{y: $0.0649$}
		  }
		  \Fdia[0.4cm]{$97.94$}_{throw}
		  
		}
		\pstree{\Frect[0.6cm]{Pour?}\nbput{neg: $0.9893$}}
		       {
			 \pstree{\Foval[0.5cm]{Inf?}^{pour}}
				{
				  \Fdia[0.4cm]{$99.94$}\naput{n: $0.999993$}
				  \Fdia[0.4cm]{$-0.06$}\nbput{y: $0.000007$}
				}
				\Fdia[0.4cm]{$97.94$}_{throw}
		       }
	}
	\pstree{\Frect[0.6cm]{Pour?}_{n}}
	{
	  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
	  {
	    \Fdia[0.4cm]{$0$}\naput{y: $0.0007$}
	    \Fdia[0.4cm]{$100$}\nbput{n: $0.9993$}
	  }
	  \Fdia[0.4cm]{$98$}_{throw}
	}

}
}
\end{center}
}
\fromSlide*{2}{

The decision tree can be solved by going from the leaves towards the root:
\begin{itemize}
\item Take \ul{\Green{weighted sum}} through chance nodes.
\item Take \ul{\Green{max}} through decision nodes.
\end{itemize}
 


}
\end{slide}}
}
{

\begin{slide}{Solving decision trees I}

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,4)

\Fnode{2}{2}{i}{Inf}
\Fnode{7}{2}{t}{Test}
\arc{i}{t}
\pspolygon[linecolor=black](0.5,0.5)(8.5,0.5)(8.5,3.5)(0.5,3.5)

%\showgrid
\end{pspicture}

\vspace*{-2.3cm}
\psset{treesep=0.25cm}
\psset{levelsep=2.5cm}
\psset{npos=0.5}
\psset{labelsep=0.5pt}
\psset{nrot=:U}

\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect{T}}
{
	\pstree{\Foval[0.5cm]{Test}^{y}}
	{
		\pstree{\Frect[0.6cm]{Pour?}\naput{pos: $0.0107$}}
		{
		  
		  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
		  {
		    \Fdia[0.4cm]{$99.94$}\naput{n: $0.9351$}
		    \Fdia[0.4cm]{$-0.06$}\nbput{y: $0.0649$}
		  }
		  \Fdia[0.4cm]{$97.94$}_{throw}
		  
		}
		\pstree{\Frect[0.6cm]{Pour?}\nbput{neg: $0.9893$}}
		       {
			 \pstree{\Foval[0.5cm]{Inf?}^{pour}}
				{
				  \Fdia[0.4cm]{$99.94$}\naput{n: $0.999993$}
				  \Fdia[0.4cm]{$-0.06$}\nbput{y: $0.000007$}
				}
				\Fdia[0.4cm]{$97.94$}_{throw}
		       }
	}
	\pstree{\Frect[0.6cm]{Pour?}_{n}}
	{
	  \pstree{\Foval[0.5cm]{Inf?}^{pour}}
	  {
	    \Fdia[0.4cm]{$0$}\naput{y: $0.0007$}
	    \Fdia[0.4cm]{$100$}\nbput{n: $0.9993$}
	  }
	  \Fdia[0.4cm]{$98$}_{throw}
	}

}
}
\end{center}

The decision tree can be solved by going from the leaves towards the root:
\begin{itemize}
\item Take \ul{\Green{weighted sum}} through chance nodes.
\item Take \ul{\Green{max}} through decision nodes.
\end{itemize}
\end{slide}



}


%####################################################################################
%# Solving decision trees II
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Solving decision trees II}

\onlySlide*{1}{

\psset{treesep=0.45cm}
\psset{levelsep=2.7cm}

\psset{npos=0.5}
\psset{labelsep=0.5pt}
\psset{nrot=:U}
\vspace*{-0.8cm}
\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}^{$d_1^1$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}\naput{$0.7$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$E$}^{$d_1^2$}}
		  {
		    \Fdia[0.4cm]{$1$}\naput{$0.4$}
		    \Fdia[0.4cm]{$2$}\nbput{$0.6$}
		  }

		  \pstree{\Foval[0.5cm]{$E$}^{$d_2^2$}}
		  {
		    \Fdia[0.4cm]{$-3$}\naput{$0.3$}
		    \Fdia[0.4cm]{$4$}\nbput{$0.7$}
		  }

		}
		  \Fdia[0.4cm]{$1.5$}\nbput{$0.3$}		
	}
	\pstree{\Foval[0.6cm]{$B$}_{$d_2^1$}}
	{
	  \Fdia[0.4cm]{$2$}\naput{$0.9$}
	  \Fdia[0.4cm]{$0$}\nbput{$0.1$}
	}
	\pstree{\Foval[0.6cm]{$C$}_{$d_3^1$}}
	{
	  \Fdia[0.4cm]{$3$}\naput{$0.8$}
	  \Fdia[0.4cm]{$-1$}\nbput{$0.2$}
	}

}
}
\end{center}
}
\onlySlide*{2}{


\psset{treesep=0.45cm}
\psset{levelsep=2.7cm}
\psset{npos=0.5}
\psset{labelsep=0.5pt}
\psset{nrot=:U}

%\hspace*{-0.8cm}
\vspace*{-0.8cm}
\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}^{$d_1^1$}_{$\Purple{1.78}$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}\naput{$0.7$}\nbput{$\Purple{1.33}$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$E$}^{$d_1^2$}_{$\Purple{1.6}$}}
		  {
		    \Fdia[0.4cm]{$1$}\naput{$0.4$}\nbput{$\Purple{0.4}$}
		    \Fdia[0.4cm]{$2$}\nbput{$0.6$}\naput{$\Purple{1.2}$}
		  }
		  \psset{linecolor=black}
		  \pstree{\Foval[0.5cm]{$E$}^{$d_2^2$}_{$\Purple{\ul{1.9}}$}}
		  {
		    \psset{linecolor=green}
		    \Fdia[0.4cm]{$-3$}\naput{$0.3$}\nbput{$\Purple{-0.9}$}
		    \Fdia[0.4cm]{$4$}\nbput{$0.7$}\naput{$\Purple{2.8}$}
		  }

		}
		  \Fdia[0.4cm]{$1.5$}\nbput{$0.3$}\naput{$\Purple{0.45}$}		
	}
	\pstree{\Foval[0.6cm]{$B$}_{$d_2^1$}^{$\Purple{1.8}$}}
	{
	  \Fdia[0.4cm]{$2$}\naput{$0.9$}\nbput{$\Purple{1.8}$}
	  \Fdia[0.4cm]{$0$}\nbput{$0.1$}\naput{$\Purple{0}$}
	}
	\psset{linecolor=black}
	\pstree{\Foval[0.6cm]{$C$}_{$d_3^1$}^{$\Purple{\ul{2.2}}$}}
	{
	  \psset{linecolor=green}
	  \Fdia[0.4cm]{$3$}\naput{$0.8$}\nbput{$\Purple{2.4}$}
	  \Fdia[0.4cm]{$-1$}\nbput{$0.2$}\naput{$\Purple{-0.2}$}
	}

}
}
\end{center}

}
\end{slide}
}
}
{
\begin{slide}{Solving decision trees II}

\psset{treesep=0.45cm}
\psset{levelsep=2.7cm}
\psset{npos=0.5}
\psset{labelsep=0.5pt}
\psset{nrot=:U}

%\hspace*{-0.8cm}
\vspace*{-0.8cm}
\begin{center}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}^{$d_1^1$}_{$\Purple{1.78}$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}\naput{$0.7$}\nbput{$\Purple{1.33}$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$E$}^{$d_1^2$}_{$\Purple{1.6}$}}
		  {
		    \Fdia[0.4cm]{$1$}\naput{$0.4$}\nbput{$\Purple{0.4}$}
		    \Fdia[0.4cm]{$2$}\nbput{$0.6$}\naput{$\Purple{1.2}$}
		  }
		  \psset{linecolor=black}
		  \pstree{\Foval[0.5cm]{$E$}^{$d_2^2$}_{$\Purple{\ul{1.9}}$}}
		  {
		    \psset{linecolor=green}
		    \Fdia[0.4cm]{$-3$}\naput{$0.3$}\nbput{$\Purple{-0.9}$}
		    \Fdia[0.4cm]{$4$}\nbput{$0.7$}\naput{$\Purple{2.8}$}
		  }

		}
		  \Fdia[0.4cm]{$1.5$}\nbput{$0.3$}\naput{$\Purple{0.45}$}		
	}
	\pstree{\Foval[0.6cm]{$B$}_{$d_2^1$}^{$\Purple{1.8}$}}
	{
	  \Fdia[0.4cm]{$2$}\naput{$0.9$}\nbput{$\Purple{1.8}$}
	  \Fdia[0.4cm]{$0$}\nbput{$0.1$}\naput{$\Purple{0}$}
	}
	\psset{linecolor=black}
	\pstree{\Foval[0.6cm]{$C$}_{$d_3^1$}^{$\Purple{\ul{2.2}}$}}
	{
	  \psset{linecolor=green}
	  \Fdia[0.4cm]{$3$}\naput{$0.8$}\nbput{$\Purple{2.4}$}
	  \Fdia[0.4cm]{$-1$}\nbput{$0.2$}\naput{$\Purple{-0.2}$}
	}

}
}
\end{center}

\end{slide}



}



%####################################################################################
%# Decision trees: characteristics
%####################################################################################


\begin{slide}{Decision trees: characteristics}
\begin{minipage}{\linewidth}
\Blue{Advantages:}
\begin{dinglist}{228}
\item All scenarios are represented explicitly.
\item Very few restrictions on the decision problems that can be represented.
\end{dinglist}
\ \\

\Red{Disadvantages:}
\begin{dinglist}{228}
\item Two separate models are used: one representing the structure and
  one representing the uncertainties.
\item The size of the decision trees grows exponentially in the number of variables.
\end{dinglist}

\end{minipage}
\end{slide}


%####################################################################################
%# A sequence of decisions
%####################################################################################


\begin{slide}{An alternative representation}



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\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,13)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}

\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

%% \rput(11,9){\rnode{e1}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

%% \rput(2,2){\rnode{e2}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

%% \rput(8,2){\rnode{e3}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}


%\showgrid
\end{pspicture}
\end{center}
\ \\
\ \\
\ \\

\centerline{But how do we represent the sequence of decisions and observations?}

\end{slide}


%####################################################################################
%# Representing decisions/observations sequences I
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Representing the decision sequence}

\onlySlide*{1}{

\ul{Possible representation:}
\vspace*{-0.4cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,14)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}


\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}

\arc[black]{mh0}{mfc}

\arc[black]{mh0}{msc}
\arc[black]{mh1}{msc}
\arc[black]{mfc}{msc}
\arc[black]{fc}{msc}

\arc[black]{mh0}{d}
\arc[black]{mh1}{d}
\arc[black]{mfc}{d}
\arc[black]{fc}{d}
\arc[black]{mh}{d}
\arc[black]{sc}{d}
\arc[black]{msc}{d}
\end{pspicture}
\end{center}
\ \\
\ \\

All nodes observed before a decision are parents of that decision.
\begin{dinglist}{228}
\item Assuming that the decision maker doesn't forget, then \Red{some links
  are redundant}!
\end{dinglist}


}
\onlySlide*{2}{

\ul{A better representation (an influence diagram):}
\vspace*{-0.4cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,14)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}


\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}

\arc[red]{mh0}{mfc}
%\arc[red]{mfc}{fc}
\arc[red]{fc}{msc}
\arc[red]{mh1}{msc}
%\arc[red]{msc}{sc}
\arc[red]{sc}{d}
\arc[red]{mh}{d}
%\arc[black]{d}{bh}




%% \rput(11,9){\rnode{e1}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

%% \rput(2,2){\rnode{e2}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

%% \rput(8,2){\rnode{e3}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}


%\showgrid
\end{pspicture}
\end{center}
\ul{\Green{Advantages:}}
\begin{itemize}
\item You can read the sequence of decisions.
\item You can read what is known at each point of decision.
\end{itemize}


}


\end{slide}

}
}
{

\begin{slide}{Specify the history}


\ul{Possible representation:}
\vspace*{-0.4cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,14)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}


\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}


\arc[black]{mh0}{mfc}

\arc[black]{mh0}{msc}
\arc[black]{mh1}{msc}
\arc[black]{mfc}{msc}
\arc[black]{fc}{msc}

\arc[black]{mh0}{d}
\arc[black]{mh1}{d}
\arc[black]{mfc}{d}
\arc[black]{fc}{d}
\arc[black]{mh}{d}
\arc[black]{sc}{d}
\arc[black]{msc}{d}




%% \rput(11,9){\rnode{e1}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

%% \rput(2,2){\rnode{e2}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

%% \rput(8,2){\rnode{e3}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}


%\showgrid
\end{pspicture}
\end{center}

\ \\
\ \\

\ul{Paths in the decision tree:}

\vspace*{-1cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(33,5)

\Fnode{2}{2}{mh0}{$\mbox{MH}_0$}
\Fbox[0.6cm]{5}{2}{mfc}{MFC}
\Fnode{8}{2}{fc}{FC}
\Fnode{11}{2}{mh1}{$\mbox{MH}_1$}
\Fbox[0.6cm]{14}{2}{msc}{MSC}
\Fnode{17}{2}{sc}{$\mbox{SC}$}
\Fnode{20}{2}{mh}{$\mbox{MH}_2$}
\Fbox{23}{2}{d}{D}
\Fnode{26}{2}{bh}{$\mbox{BH}$}
\Fdiam{24.5}{5}{u}{U}



\arc{mh0}{mfc}
\arc{mfc}{fc}
\arc{fc}{mh1}
\arc{mh1}{msc}
\arc{msc}{sc}
\arc{sc}{mh}
\arc{mh}{d}
\arc{d}{bh}
\arc{d}{u}
\arc{bh}{u}




%\showgrid
\end{pspicture}
\end{center}
\end{slide}

\begin{slide}{Representing decisions/observations sequences II}

\ul{A better representation (an influence diagram):}
\vspace*{-0.4cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,14)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}


\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}

\arc[red]{mh0}{mfc}
%\arc[red]{mfc}{fc}
\arc[red]{fc}{msc}
\arc[red]{mh1}{msc}
%\arc[red]{msc}{sc}
\arc[red]{sc}{d}
\arc[red]{mh}{d}
%\arc[black]{d}{bh}




%% \rput(11,9){\rnode{e1}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

%% \rput(2,2){\rnode{e2}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

%% \rput(8,2){\rnode{e3}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}


%\showgrid
\end{pspicture}
\end{center}
\ul{\Green{Advantages:}}
\begin{itemize}
\item You can read the sequence of decisions.
\item You can read what is known at each point of decision.
\end{itemize}


\end{slide}



}


%####################################################################################
%# Influence diagrams
%####################################################################################


\begin{slide}{Influence diagrams}



\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\hspace*{-0.5cm}
\begin{pspicture}(1,1)(30,11)

\Fbox{2}{5}{d1}{$D_1$}
\Fdiam{5}{2}{u1}{$U_1$}
\Fnode{5}{7}{b}{$B$}
\Fnode{11}{7}{d}{$D$}
\Fnode{17}{7}{f}{$F$}
\Fnode{23}{7}{h}{$H$}
\Fnode{29}{7}{i}{$I$}

\Fnode{2}{10}{a}{$A$}
\Fnode{8}{10}{c}{$C$}
\Fnode{14}{10}{e}{$E$}
\Fnode{20}{10}{g}{$G$}
\Fnode{26}{10}{k}{$K$}
\Fnode{26}{5}{j}{$J$}

\Fbox{9}{4}{d2}{$D_2$}
\Fbox{19}{4}{d3}{$D_3$}
\Fbox{26}{2}{d4}{$D_4$}

\Fdiam{13}{2}{u2}{$U_2$}
\Fdiam{22}{2}{u3}{$U_3$}
\Fdiam{30}{3}{u4}{$U_4$}

\arc[red]{a}{d1}
\arc{a}{b}
\arc{d1}{b}
\arc[green]{d1}{u1}
\arc[green]{c}{u1}
\arc{b}{c}
\arc{b}{d}
\arc{d}{e}
\arc{c}{e}
\arc{e}{f}
\arc{d2}{f}
\arc[red]{f}{d3}
\arc{f}{g}
\arc{g}{k}
\arc{g}{h}
\arc{h}{j}
\arc{i}{j}
\arc{i}{k}
\arc[green]{k}{u4}
\arc[green]{d4}{u4}
\arc{d3}{j}
\arc[green]{d3}{u3}
\arc[green]{j}{u3}
\arc[red]{j}{d4}
\arc[green]{d2}{u2}
\arc[red]{c}{d2}

%\showgrid
\end{pspicture}
\ul{Nodes and links:} 
\begin{itemize}
\item[] $\Var{\bigcirc}$ Chance variable $\Green{\rightarrow}$ \Green{causal links}
\item[] $\Red{\Box}$ Decision variable $\Red{\rightarrow}$ \Red{information
  links}
\item[] $\Purple{\Diamond}$ Utility function $\Green{\rightarrow}$
  \Purple{utility link}, $U=\sum _iU_i$.
\end{itemize}
\ul{Note:}
\begin{itemize}
\item We assume no-forgetting.
\item A directed path comprising all decisions $\Rightarrow$ the
  scenario is well-defined.
\end{itemize}



\end{slide}



%####################################################################################
%# Influence diagrams and Hugin
%####################################################################################


\begin{slide}{Influence diagrams and Hugin}

In Hugin the nodes are depicted as:
\begin{itemize}
\item[] Chance variable $\bigcirc$
\item[] Decision variable: $\Box$
\item[] Utility nodes: $\Diamond$
\end{itemize}
\ \\
\ \\

\ul{\Green{Note that:}}
\begin{itemize}
\item No tables are specified for decision nodes.
\item A utility function is specified for a utility node.
\end{itemize}



\end{slide}




%####################################################################################
%# Influence diagrams: Characteristics
%####################################################################################


\begin{slide}{Influence diagrams: Characteristics}
\begin{minipage}{\linewidth}
\Blue{Advantages:}
\begin{dinglist}{228}
\item Grows only linearly in the number of variables.
\item Requires only one model for representing both structure as well
  as the uncertainty model.
\end{dinglist}
\ \\
\ \\

\Red{Disadvantages:}
\begin{dinglist}{228}
\item The sequence of observations and decisions is the same in all
  scenarios (the decision problem is symmetric).
\end{dinglist}
\ \\
\ \\

\Green{Definition:} A decision problem is said to be symmetric if:
\begin{dinglist}{228}
\item In all decision tree representations, the number of scenarios is
  the same as the cardinality of the Cartesian product of the state
  spaces of all chance and decision variables.
\item in one decision tree representation, the sequence of
  observations and decisions is the same in all scenarios.
\end{dinglist}




\end{minipage}
\end{slide}




%####################################################################################
%# Symmetric decision trees 
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Symmetric decision trees}



\fromSlide*{1}{

\vspace*{-0.7cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,5.5)

\Fbox[0.2cm]{1}{1}{d1}{$D_1$}
\Fbox[0.2cm]{6}{1}{d2}{$D_2$}
\Fnode[0.2cm]{4}{2}{b}{$B$}
\Fnode[0.2cm]{1}{3}{f}{$F$}
\Fnode[0.2cm]{3}{4}{a}{$A$}
\Fnode[0.2cm]{6}{4}{c}{$C$}
\Fnode[0.2cm]{9}{1}{e}{$E$}
\Fdiam[0.2cm]{9}{4}{u}{$U$}

\arc{d1}{b}
\arc{f}{b}
\arc{c}{b}
\arc{a}{c}
\arc{c}{u}
\arc{c}{e}
\arc{d2}{c}

\pspolygon[linecolor=black](0,0)(10.5,0)(10.5,5.2)(0,5.2)


\end{pspicture}


\psset{treesep=0.05cm}
\psset{levelsep=2cm}

\psset{npos=0.5}
\psset{treefit=loose}
\psset{labelsep=0.5pt}
\psset{nrot=:U}
\vspace*{-2.5cm}
\begin{center}
\hspace*{2.5cm}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}




}
}
\end{center}
}
\fromSlide*{2}{

%\vspace*{-0.4cm}
\ul{The sequence of observations and decisions is the same in all scenarios:}
\vspace*{-0.3cm}
\begin{center}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,2.5)

\Fbox[0.2cm]{1}{1}{d1}{$D_1$}
\Fnode[0.2cm]{4}{1}{a}{$A$}
\Fbox[0.2cm]{7}{1}{d2}{$D_2$}
\Fnode[0.2cm]{10}{1}{c}{$C$}



\arc{d1}{a}
\arc{a}{d2}
\arc{d2}{c}


\end{pspicture}
\end{center}

}
\end{slide}
}
}
{

\begin{slide}{Symmetric decision trees}


\vspace*{-0.7cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,5.5)

\Fbox[0.2cm]{1}{1}{d1}{$D_1$}
\Fbox[0.2cm]{6}{1}{d2}{$D_2$}
\Fnode[0.2cm]{4}{2}{b}{$B$}
\Fnode[0.2cm]{1}{3}{f}{$F$}
\Fnode[0.2cm]{3}{4}{a}{$A$}
\Fnode[0.2cm]{6}{4}{c}{$C$}
\Fnode[0.2cm]{9}{1}{e}{$E$}
\Fdiam[0.2cm]{9}{4}{u}{$U$}

\arc{d1}{b}
\arc{f}{b}
\arc{c}{b}
\arc{a}{c}
\arc{c}{u}
\arc{c}{e}
\arc{d2}{c}

\pspolygon[linecolor=black](0,0)(10.5,0)(10.5,5.2)(0,5.2)


\end{pspicture}


\psset{treesep=0.05cm}
\psset{levelsep=2cm}

\psset{npos=0.5}
\psset{treefit=loose}
\psset{labelsep=0.5pt}
\psset{nrot=:U}
\vspace*{-2.5cm}
\begin{center}
\hspace*{2.5cm}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}



}}

\end{center}



\ul{The sequence of observations and decisions is the same in all
  scenarios:}
\vspace*{-0.3cm}
\begin{center}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,2.5)

\Fbox[0.2cm]{1}{1}{d1}{$D_1$}
\Fnode[0.2cm]{4}{1}{a}{$A$}
\Fbox[0.2cm]{7}{1}{d2}{$D_2$}
\Fnode[0.2cm]{10}{1}{c}{$C$}



\arc{d1}{a}
\arc{a}{d2}
\arc{d2}{c}


\end{pspicture}
\end{center}


\end{slide}

}


%####################################################################################
%# Optimal strategy I
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Optimal strategy I}

\fromSlide*{1}{
\vspace*{-0.7cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,5.5)

\Fbox[0.2cm]{1}{1}{d1}{$D_1$}
\Fbox[0.2cm]{6}{1}{d2}{$D_2$}
\Fnode[0.2cm]{3.5}{2}{b}{$B$}
\Fnode[0.2cm]{1}{3}{f}{$F$}
\Fnode[0.2cm]{3}{4}{a}{$A$}
\Fnode[0.2cm]{6}{4}{c}{$C$}
\Fnode[0.2cm]{9}{1}{e}{$E$}
\Fdiam[0.2cm]{9}{4}{u}{$U$}

\arc{d1}{b}
\arc{f}{b}
\arc{c}{b}
\arc{a}{c}
\arc{c}{u}
\arc{c}{e}
\arc{d2}{c}
\arc[red]{d1}{d2}
\arc[red]{a}{d2}

\pspolygon[linecolor=black](0,0)(10.5,0)(10.5,5.2)(0,5.2)


\end{pspicture}


\psset{treesep=0.05cm}
\psset{levelsep=1.4cm}

\psset{npos=0.5}
\psset{treefit=loose}
\psset{labelsep=0.5pt}
\psset{nrot=:U}
\vspace*{-2.5cm}
\begin{center}
\hspace*{2.5cm}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}




}
}
\end{center}
}
\fromSlide*{2}{
\vspace*{-1.2cm}
\ul{\Green{Solution for influene diagrams:}}
\begin{enumerate}
\item Determine a \ul{policy} for $\Red{D_2}$: $\sigma
  _{\Red{D_2}}(\Red{D_1},\Var{A})\rightarrow \Red{D_2}$.\newline
For this we need $P(\Var{C}|\Red{D_1},\Var{A},\Red{D_2})$.
\item Use $\sigma _{\Red{D_2}}$ for determining a \ul{policy} for $\Red{D_1}$: $\sigma
  _{\Red{D_1}}\rightarrow \Red{D_1}$.\newline
For this we need $P(\Var{A}|\Red{D_1})$.
\end{enumerate}
All probabilities can be achieved from the model without folding out
the decision tree.
}

\end{slide}
}
}
{

\begin{slide}{Optimal strategy I}

\vspace*{-0.7cm}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{pspicture}(0,0)(10,5.5)

\Fbox[0.2cm]{1}{1}{d1}{$D_1$}
\Fbox[0.2cm]{6}{1}{d2}{$D_2$}
\Fnode[0.2cm]{3.5}{2}{b}{$B$}
\Fnode[0.2cm]{1}{3}{f}{$F$}
\Fnode[0.2cm]{3}{4}{a}{$A$}
\Fnode[0.2cm]{6}{4}{c}{$C$}
\Fnode[0.2cm]{9}{1}{e}{$E$}
\Fdiam[0.2cm]{9}{4}{u}{$U$}

\arc{d1}{b}
\arc{f}{b}
\arc{c}{b}
\arc{a}{c}
\arc{c}{u}
\arc{c}{e}
\arc{d2}{c}
\arc[red]{d1}{d2}
\arc[red]{a}{d2}

\pspolygon[linecolor=black](0,0)(10.5,0)(10.5,5.2)(0,5.2)


\end{pspicture}


\psset{treesep=0.05cm}
\psset{levelsep=1.4cm}

\psset{npos=0.5}
\psset{treefit=loose}
\psset{labelsep=0.5pt}
\psset{nrot=:U}
\vspace*{-2.5cm}
\begin{center}
\hspace*{2.5cm}
\mbox{
\pstree[treemode=R,arrows=->,linecolor=green]{\Frect[0.5cm]{$D_1$}}
{
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}
	\pstree{\Foval[0.5cm]{$A$}}
	{
		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}


		\pstree{\Frect[0.6cm]{$D_2$}}
		{
		  
		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		  \pstree{\Foval[0.5cm]{$C$}}
		  {
		    \Fdia[0.4cm]{}
		    \Fdia[0.4cm]{}
		  }

		}
	}




}
}
\end{center}

\vspace*{-1.2cm}
\ul{\Green{Solution for influene diagrams:}}
\begin{enumerate}
\item Determine a \ul{policy} for $\Red{D_2}$: $\sigma
  _{\Red{D_2}}(\Red{D_1},\Var{A})\rightarrow \Red{D_2}$.\newline
For this we need $P(\Var{C}|\Red{D_1},\Var{A},\Red{D_2})$.
\item Use $\sigma _{\Red{D_2}}$ for determining a \ul{policy} for $\Red{D_1}$: $\sigma
  _{\Red{D_1}}\rightarrow \Red{D_1}$.\newline
For this we need $P(\Var{A}|\Red{D_1})$.
\end{enumerate}
All probabilities can be achieved from the model without folding out
the decision tree.

\end{slide}


}

%####################################################################################
%# Optimal strategy II
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Optimal strategy II}

\fromSlide*{1}{
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,14)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}


\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}

\arc[red]{mh0}{mfc}
%\arc[red]{mfc}{fc}
\arc[red]{fc}{msc}
\arc[red]{mh1}{msc}
%\arc[red]{msc}{sc}
\arc[red]{sc}{d}
\arc[red]{mh}{d}
%\arc[black]{d}{bh}




%% \rput(11,9){\rnode{e1}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

%% \rput(2,2){\rnode{e2}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

%% \rput(8,2){\rnode{e3}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}


%\showgrid
\end{pspicture}
\end{center}
}
\onlySlide*{1}{
\ul{The policy for $\DecM{D}$:}
$\sigma
_{\DecM{D}}(\VarM{MH}_{\Var{0}},\DecM{MFC},\VarM{FC},\VarM{MH}_{\Var{1}},\DecM{MSC},\VarM{SC},\VarM{MH}_{\Var{2}})\rightarrow
\DecM{D} $\newline\newline
\ul{We request:}
$
P(\VarM{BH}|\VarM{MH}_{\Var{0}},\DecM{MFC},\VarM{FC},\VarM{MH}_{\Var{1}},\DecM{MSC},\VarM{SC},\VarM{MH}_{\Var{2}},\DecM{D})
$\newline\newline


}

\onlySlide*{2}{
\ul{The policy for $\DecM{D}$:}
$\sigma
_{\DecM{D}}(\mbox{MH}_0,\mbox{MFC},\Green{\ul{\mbox{FC}}},\mbox{MH}_1,\mbox{MSC},\Green{\ul{\mbox{SC}}},
\Green{\ul{\mbox{MH}_2}})\rightarrow \DecM{D}
$\newline\newline
\ul{We request:}
$
P(\VarM{BH}|\mbox{MH}_0,\mbox{MFC},\Green{\ul{\mbox{FC}}},\mbox{MH}_1,\mbox{MSC},\Green{\ul{\mbox{SC}}},
\Green{\ul{\mbox{MH}_2}},\Green{\ul{\mbox{D}}})
$\newline\newline

\centerline{
\Green{From d-separation we can find the \ul{relevant past}!}
}
}


\end{slide}
}
}
{

\begin{slide}{Optimal strategy II}


\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\begin{center}
\begin{pspicture}(1,1)(22,14)

\Fnode{2}{5}{oh0}{$\mbox{OH}_0$}
\Fnode{5}{2}{fc}{FC}
\Fnode{8}{5}{oh1}{$\mbox{OH}_1$}
\Fnode{11}{2}{sc}{$\mbox{SC}$}
\Fnode{14}{5}{oh}{$\mbox{OH}_2$}
\Fnode{18}{7}{bh}{$\mbox{BH}$}
\Fnode{14}{9}{mh}{$\mbox{MH}_2$}
\Fnode{8}{9}{mh1}{$\mbox{MH}_1$}
\Fnode{2}{9}{mh0}{$\mbox{MH}_0$}


\Fdiam{21}{5}{u}{U}
\arc{bh}{u}

\Fbox{18}{3}{d}{D}
\arc{d}{u}

\Fbox[0.6cm]{11}{12}{msc}{MSC}
\Fbox[0.6cm]{5}{12}{mfc}{MFC}

\arc{mfc}{mh1}
\arc{msc}{mh}

\arc{oh0}{fc}
\arc{oh0}{oh1}
\arc{fc}{oh1}
\arc{oh1}{oh}
\arc{oh1}{sc}
\arc{sc}{oh}
\arc{oh}{bh}
\arc{mh}{bh}
\arc{mh1}{mh}
\arc{mh0}{mh1}

\arc[red]{mh0}{mfc}
%\arc[red]{mfc}{fc}
\arc[red]{fc}{msc}
\arc[red]{mh1}{msc}
%\arc[red]{msc}{sc}
\arc[red]{sc}{d}
\arc[red]{mh}{d}
%\arc[black]{d}{bh}




%% \rput(11,9){\rnode{e1}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e1}{mh}

%% \rput(2,2){\rnode{e2}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e2}{fc}

%% \rput(8,2){\rnode{e3}{}}
%% \ncline[linestyle=dotted,arrows=->,linecolor=red]{e3}{sc}


%\showgrid
\end{pspicture}
\end{center}


\ul{The policy for $\DecM{D}$:}
$\sigma
_{\DecM{D}}(\mbox{MH}_0,\mbox{MFC},\Green{\ul{\mbox{FC}}},\mbox{MH}_1,\mbox{MSC},\Green{\ul{\mbox{SC}}},
\Green{\ul{\mbox{MH}_2}})\rightarrow \DecM{D}
$\newline\newline
\ul{We request:}
$
P(\VarM{BH}|\mbox{MH}_0,\mbox{MFC},\Green{\ul{\mbox{FC}}},\mbox{MH}_1,\mbox{MSC},\Green{\ul{\mbox{SC}}},
\Green{\ul{\mbox{MH}_2}},\Green{\ul{\mbox{D}}})
$\newline\newline

\centerline{
\Green{From d-separation we can find the \ul{relevant past}!}
}



\end{slide}

}





%####################################################################################
%# Fishing in the north sea
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{Fishing in the north sea}

\fromSlide*{1}{


Based on measurements, $\Var{T}$, a quota for fishing volume,
$\Var{FV}$, for next year is decided. The amount of fish, $\Var{V}$,
and the quota determines the utility.
\ \\
\ \\


}
\fromSlide*{2}{

\ul{A five year period:}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\vspace*{-0.3cm}
\begin{center}
\begin{pspicture}(6,0)(35,7)


\Fnode{10}{5}{t1}{$T_1$}
\Fbox{10}{2}{fv1}{$FV_1$}
\Fnode{7}{5}{v1}{$V_1$}
\Fdiam{7}{2}{u1}{$U_1$}

\arc{v1}{t1}
\arc[red]{t1}{fv1}
\arc{fv1}{u1}
\arc{v1}{u1}


\Fnode{16}{5}{t2}{$T_2$}
\Fbox{16}{2}{fv2}{$FV_2$}
\Fnode{13}{5}{v2}{$V_2$}
\Fdiam{13}{2}{u2}{$U_2$}

\arc{v2}{t2}
\arc[red]{t2}{fv2}
\arc{fv2}{u2}
\arc{v2}{u2}


\Fnode{22}{5}{t3}{$T_3$}
\Fbox{22}{2}{fv3}{$FV_3$}
\Fnode{19}{5}{v3}{$V_3$}
\Fdiam{19}{2}{u3}{$U_3$}

\arc{v3}{t3}
\arc[red]{t3}{fv3}
\arc{fv3}{u3}
\arc{v3}{u3}


\Fnode{28}{5}{t4}{$T_4$}
\Fbox{28}{2}{fv4}{$FV_4$}
\Fnode{25}{5}{v4}{$V_4$}
\Fdiam{25}{2}{u4}{$U_4$}

\arc{v4}{t4}
\arc[red]{t4}{fv4}
\arc{fv4}{u4}
\arc{v4}{u4}


\Fnode{34}{5}{t5}{$T_5$}
\Fbox{34}{2}{fv5}{$FV_5$}
\Fnode{31}{5}{v5}{$V_5$}
\Fdiam{31}{2}{u5}{$U_5$}

\arc{v5}{t5}
\arc[red]{t5}{fv5}
\arc{fv5}{u5}
\arc{v5}{u5}

\arc{fv1}{v2}
\arc{fv2}{v3}
\arc{fv3}{v4}
\arc{fv4}{v5}

\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v1}{v2}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v2}{v3}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v3}{v4}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v4}{v5}
%\showgrid
\end{pspicture}
\end{center}

Unfortunately, the optimal policy for $\Red{FV_5}$ depends on the
entire past:
\[
\sigma
_{\Red{FV_5}}(\Var{T_1},\Red{FV_1},\Var{T_2},\Red{FV_2},\Var{T_3},\Red{FV_3},\Var{T_4},\Red{FV_4},\Var{T_5})
\]
\centerline{\Red{This
  is intractable!}}

}
\end{slide}
}
}
{

\begin{slide}{Fishing in the north sea}


\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(4,0)(30,7)


\Fnode{12}{5}{t}{$T$}
\Fbox{12}{2}{fv}{$FV$}
\Fnode{7}{5}{v}{$V$}
\Fdiam{7}{2}{u}{$U$}

\arc{v}{t}
\arc[red]{t}{fv}
\arc{fv}{u}
\arc{v}{u}
\ncline[doubleline=true,arrows=->,linecolor=black]{fv}{v}


\pspolygon[linecolor=black](14.5,4.5)(14.5,5.5)(15.5,5.5)(15.5,4.5)
\rput[c](15,5){$5$}

\psline[doubleline=true,arrows=->,linecolor=black](7,5.8)(7,7)(5,7)(5,5)(6.2,5)

\rput[l](18,5){Based on measurements, $\Var{T}$, a quota for}
\rput[l](18,4){fishing volume, $\Var{FV}$, for next year is decided.}
\rput[l](18,3){The amount of fish, $\Var{V}$, and the quota}
\rput[l](18,2){determines the utility.}

%\showgrid
\end{pspicture}
\end{center}

\ul{The unfolded version:}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\vspace*{-0.3cm}
\begin{center}
\begin{pspicture}(6,0)(35,7)


\Fnode{10}{5}{t1}{$T_1$}
\Fbox{10}{2}{fv1}{$FV_1$}
\Fnode{7}{5}{v1}{$V_1$}
\Fdiam{7}{2}{u1}{$U_1$}

\arc{v1}{t1}
\arc[red]{t1}{fv1}
\arc{fv1}{u1}
\arc{v1}{u1}


\Fnode{16}{5}{t2}{$T_2$}
\Fbox{16}{2}{fv2}{$FV_2$}
\Fnode{13}{5}{v2}{$V_2$}
\Fdiam{13}{2}{u2}{$U_2$}

\arc{v2}{t2}
\arc[red]{t2}{fv2}
\arc{fv2}{u2}
\arc{v2}{u2}


\Fnode{22}{5}{t3}{$T_3$}
\Fbox{22}{2}{fv3}{$FV_3$}
\Fnode{19}{5}{v3}{$V_3$}
\Fdiam{19}{2}{u3}{$U_3$}

\arc{v3}{t3}
\arc[red]{t3}{fv3}
\arc{fv3}{u3}
\arc{v3}{u3}


\Fnode{28}{5}{t4}{$T_4$}
\Fbox{28}{2}{fv4}{$FV_4$}
\Fnode{25}{5}{v4}{$V_4$}
\Fdiam{25}{2}{u4}{$U_4$}

\arc{v4}{t4}
\arc[red]{t4}{fv4}
\arc{fv4}{u4}
\arc{v4}{u4}


\Fnode{34}{5}{t5}{$T_5$}
\Fbox{34}{2}{fv5}{$FV_5$}
\Fnode{31}{5}{v5}{$V_5$}
\Fdiam{31}{2}{u5}{$U_5$}

\arc{v5}{t5}
\arc[red]{t5}{fv5}
\arc{fv5}{u5}
\arc{v5}{u5}

\arc{fv1}{v2}
\arc{fv2}{v3}
\arc{fv3}{v4}
\arc{fv4}{v5}

\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v1}{v2}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v2}{v3}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v3}{v4}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v4}{v5}
%\showgrid
\end{pspicture}
\end{center}

Unfortunately, the optimal policy for $\Red{FV_5}$ depends on the
entire past:
\[
\sigma
_{\Red{FV_5}}(\Var{T_1},\Red{FV_1},\Var{T_2},\Red{FV_2},\Var{T_3},\Red{FV_3},\Var{T_4},\Red{FV_4},\Var{T_5})\hspace*{1cm}\text{\ul{This
  is intractable!}}
\]

\end{slide}

}


%####################################################################################
%# Information blocking
%####################################################################################

\PDForPS{
\begin{slide}{Information blocking}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\vspace*{-0.3cm}
\begin{center}
\begin{pspicture}(6,0)(35,7)


\Fnode{10}{5}{t1}{$T_1$}
\Fbox{10}{2}{fv1}{$FV_1$}
\Fnode{7}{5}{v1}{$V_1$}
\Fdiam{7}{2}{u1}{$U_1$}

\arc{v1}{t1}
\arc[red]{t1}{fv1}
\arc{fv1}{u1}
\arc{v1}{u1}


\Fnode{16}{5}{t2}{$T_2$}
\Fbox{16}{2}{fv2}{$FV_2$}
\Fnode{13}{5}{v2}{$V_2$}
\Fdiam{13}{2}{u2}{$U_2$}

\arc{v2}{t2}
\arc[red]{t2}{fv2}
\arc{fv2}{u2}
\arc{v2}{u2}


\Fnode{22}{5}{t3}{$T_3$}
\Fbox{22}{2}{fv3}{$FV_3$}
\Fnode{19}{5}{v3}{$V_3$}
\Fdiam{19}{2}{u3}{$U_3$}

\arc{v3}{t3}
\arc[red]{t3}{fv3}
\arc{fv3}{u3}
\arc{v3}{u3}


\Fnode{28}{5}{t4}{$T_4$}
\Fbox{28}{2}{fv4}{$FV_4$}
\Fnode{25}{5}{v4}{$V_4$}
\Fdiam{25}{2}{u4}{$U_4$}

\arc{v4}{t4}
\arc[red]{t4}{fv4}
\arc{fv4}{u4}
\arc{v4}{u4}


\Fnode{34}{5}{t5}{$T_5$}
\Fbox{34}{2}{fv5}{$FV_5$}
\Fnode{31}{5}{v5}{$V_5$}
\Fdiam{31}{2}{u5}{$U_5$}

\arc{v5}{t5}
\arc[red]{t5}{fv5}
\arc{fv5}{u5}
\arc{v5}{u5}

\arc{fv1}{v2}
\arc{fv2}{v3}
\arc{fv3}{v4}
\arc{fv4}{v5}

\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v1}{v2}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v2}{v3}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v3}{v4}
\ncarc[arrows=->,linecolor=green,arcangleA=40,arcangleB=40]{v4}{v5}
%\showgrid
\end{pspicture}
\end{center}

To make the calculations tractable we use an approximation instead:

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(6,0)(35,7)


\Fnode{10}{5}{t1}{$T_1$}
\Fbox{10}{2}{fv1}{$FV_1$}
\Fnode{7}{5}{v1}{$V_1$}
\Fdiam{7}{2}{u1}{$U_1$}

\arc{v1}{t1}
\arc[red]{t1}{fv1}
\arc{fv1}{u1}
\arc{v1}{u1}


\Fnode{16}{5}{t2}{$T_2$}
\Fbox{16}{2}{fv2}{$FV_2$}
\Fnode{13}{5}{v2}{$V_2$}
\Fdiam{13}{2}{u2}{$U_2$}

\arc{v2}{t2}
\arc[red]{t2}{fv2}
\arc{fv2}{u2}
\arc{v2}{u2}


\Fnode{22}{5}{t3}{$T_3$}
\Fbox{22}{2}{fv3}{$FV_3$}
\Fnode{19}{5}{v3}{$V_3$}
\Fdiam{19}{2}{u3}{$U_3$}

\arc{v3}{t3}
\arc[red]{t3}{fv3}
\arc{fv3}{u3}
\arc{v3}{u3}


\Fnode{28}{5}{t4}{$T_4$}
\Fbox{28}{2}{fv4}{$FV_4$}
\Fnode{25}{5}{v4}{$V_4$}
\Fdiam{25}{2}{u4}{$U_4$}

\arc{v4}{t4}
\arc[red]{t4}{fv4}
\arc{fv4}{u4}
\arc{v4}{u4}


\Fnode{34}{5}{t5}{$T_5$}
\Fbox{34}{2}{fv5}{$FV_5$}
\Fnode{31}{5}{v5}{$V_5$}
\Fdiam{31}{2}{u5}{$U_5$}

\arc{v5}{t5}
\arc[red]{t5}{fv5}
\arc{fv5}{u5}
\arc{v5}{u5}

\arc{fv1}{v2}
\arc{t1}{v2}
\arc{fv2}{v3}
\arc{t2}{v3}
\arc{fv3}{v4}
\arc{t3}{v4}
\arc{fv4}{v5}
\arc{t4}{v5}


%\showgrid
\end{pspicture}
\end{center}
\centerline{
The probability $P(\Var{V_2}|\Var{T_1},\Red{FV_1})$ is taken from the
initial model.
}
\end{slide}
}
{

\begin{slide}{Information blocking}

To make the calculations tractable we use an approximation instead:
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(4,0)(30,7)


\Fnode{12}{5}{t}{$T$}
\Fbox{12}{2}{fv}{$FV$}
\Fnode{7}{5}{v}{$V$}
\Fdiam{7}{2}{u}{$U$}

\arc{v}{t}
\arc[red]{t}{fv}
\arc{fv}{u}
\arc{v}{u}
\ncline[doubleline=true,arrows=->,linecolor=black]{fv}{v}


\ncarc[arrows=->,linecolor=black,arcangleA=-40,arcangleB=-40,doubleline=true]{t}{v}

\pspolygon[linecolor=black](14.5,4.5)(14.5,5.5)(15.5,5.5)(15.5,4.5)
\rput[c](15,5){$5$}

%\psline[doubleline=true,arrows=->,linecolor=black](7,5.8)(7,7)(5,7)(5,5)(6.2,5)

%\showgrid
\end{pspicture}
\end{center}

\ul{The unfolded version:}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(6,0)(35,7)


\Fnode{10}{5}{t1}{$T_1$}
\Fbox{10}{2}{fv1}{$FV_1$}
\Fnode{7}{5}{v1}{$V_1$}
\Fdiam{7}{2}{u1}{$U_1$}

\arc{v1}{t1}
\arc[red]{t1}{fv1}
\arc{fv1}{u1}
\arc{v1}{u1}


\Fnode{16}{5}{t2}{$T_2$}
\Fbox{16}{2}{fv2}{$FV_2$}
\Fnode{13}{5}{v2}{$V_2$}
\Fdiam{13}{2}{u2}{$U_2$}

\arc{v2}{t2}
\arc[red]{t2}{fv2}
\arc{fv2}{u2}
\arc{v2}{u2}


\Fnode{22}{5}{t3}{$T_3$}
\Fbox{22}{2}{fv3}{$FV_3$}
\Fnode{19}{5}{v3}{$V_3$}
\Fdiam{19}{2}{u3}{$U_3$}

\arc{v3}{t3}
\arc[red]{t3}{fv3}
\arc{fv3}{u3}
\arc{v3}{u3}


\Fnode{28}{5}{t4}{$T_4$}
\Fbox{28}{2}{fv4}{$FV_4$}
\Fnode{25}{5}{v4}{$V_4$}
\Fdiam{25}{2}{u4}{$U_4$}

\arc{v4}{t4}
\arc[red]{t4}{fv4}
\arc{fv4}{u4}
\arc{v4}{u4}


\Fnode{34}{5}{t5}{$T_5$}
\Fbox{34}{2}{fv5}{$FV_5$}
\Fnode{31}{5}{v5}{$V_5$}
\Fdiam{31}{2}{u5}{$U_5$}

\arc{v5}{t5}
\arc[red]{t5}{fv5}
\arc{fv5}{u5}
\arc{v5}{u5}

\arc{fv1}{v2}
\arc{t1}{v2}
\arc{fv2}{v3}
\arc{t2}{v3}
\arc{fv3}{v4}
\arc{t3}{v4}
\arc{fv4}{v5}
\arc{t4}{v5}


%\showgrid
\end{pspicture}
\end{center}
\centerline{
The probability $P(\Var{V_2}|\Var{T_1},\Red{FV_1})$ is taken from the
initial model.
}

\end{slide}

}


%####################################################################################
%# The dangers of non-observed nodes
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{The dangers of non-observed nodes}
\fromSlide*{1}{
\ul{Temporal links between non-observed nodes are dangerous!}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\hspace*{-0.6cm}
\begin{pspicture}(6,-2)(36,7)


\Fnode{11}{5}{b1}{$B_1$}
\Fnode{11}{2}{e1}{$E_1$}
\Fnode{7}{5}{a1}{$A_1$}
\Fnode{7}{2}{c1}{$C_1$}
\Fbox{9}{-1}{d1}{$D_1$}
\Fdiam{14}{2}{u1}{$U_1$}

\arc{a1}{c1}
\arc{b1}{c1}
\arc{a1}{e1}
\arc[red]{c1}{d1}
\arc[red]{e1}{d1}
\arc{d1}{u1}
\arc{b1}{u1}


\Fnode{22}{5}{b2}{$B_2$}
\Fnode{22}{2}{e2}{$E_2$}
\Fnode{18}{5}{a2}{$A_2$}
\Fnode{18}{2}{c2}{$C_2$}
\Fbox{20}{-1}{d2}{$D_2$}
\Fdiam{25}{2}{u2}{$U_2$}

\arc{a2}{c2}
\arc{b2}{c2}
\arc{a2}{e2}
\arc[red]{c2}{d2}
\arc[red]{e2}{d2}
\arc{d2}{u2}
\arc{b2}{u2}

\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a1}{a2}
\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a1}{b2}



\Fnode{33}{5}{b3}{$B_3$}
\Fnode{33}{2}{e3}{$E_3$}
\Fnode{29}{5}{a3}{$A_3$}
\Fnode{29}{2}{c3}{$C_3$}
\Fbox{31}{-1}{d3}{$D_3$}
\Fdiam{36}{2}{u3}{$U_3$}

\arc{a3}{c3}
\arc{b3}{c3}
\arc{a3}{e3}
\arc[red]{c3}{d3}
\arc[red]{e3}{d3}
\arc{d3}{u3}
\arc{b3}{u3}

\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a2}{a3}
\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a2}{b3}



%\showgrid
\end{pspicture}
}
\fromSlide*{2}{
\ul{We introduce history variables to summarize the past:}

\vspace*{0.2cm}

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\vspace*{0cm}
\begin{center}
\begin{pspicture}(6,-2)(30,6)


\Fnode{11}{5}{b1}{$B_1$}
\Fnode{11}{2}{e1}{$E_1$}
\Fnode{7}{5}{a1}{$A_1$}
\Fnode{7}{2}{c1}{$C_1$}
\Fbox{9}{-1}{d1}{$D_1$}
\Fovalnode{15}{5}{h1}{$Hist_1$}

\Fdiam{15}{0}{u1}{$U_1$}

\arc{a1}{c1}
\arc{b1}{c1}
\arc{a1}{e1}
\arc[red]{c1}{d1}
\arc[red]{e1}{d1}
\arc{d1}{u1}
\arc{b1}{u1}
\arc{e1}{h1}
\arc{c1}{h1}


\Fnode{24}{5}{b2}{$B_2$}
\Fnode{24}{2}{e2}{$E_2$}
\Fnode{20}{5}{a2}{$A_2$}
\Fnode{20}{2}{c2}{$C_2$}
\Fbox{22}{-1}{d2}{$D_2$}
\Fovalnode{28}{5}{h2}{$Hist_2$}

\Fdiam{28}{0}{u2}{$U_2$}

\arc{a2}{c2}
\arc{b2}{c2}
\arc{a2}{e2}
\arc[red]{c2}{d2}
\arc[red]{e2}{d2}
\arc{d2}{u2}
\arc{b2}{u2}
\arc{e2}{h2}
\arc{c2}{h2}

\arc[red]{h1}{d2}

\arc{h1}{a2}
\ncarc[linecolor=green,arrows=->]{h1}{b2}
\ncarc[linecolor=green,arrows=->]{h1}{h2}

%\showgrid
\end{pspicture}
\end{center}
}
\end{slide}
}}
{
\begin{slide}{The dangers of non-observed nodes}

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\vspace*{-0.3cm}
\begin{center}
\begin{pspicture}(4,-2)(32,7)


\Fnode{11}{5}{b}{$B$}
\Fnode{11}{2}{e}{$E$}
\Fnode{7}{5}{a}{$A$}
\Fnode{7}{2}{c}{$C$}
\Fbox{9}{-1}{d}{$D$}

\Fdiam{15}{2}{u}{$U$}

\arc{a}{c}
\arc{b}{c}
\arc{a}{e}
\ncline[doubleline=true,arrows=->,linecolor=black]{a}{b}
\arc[red]{c}{d}
\arc[red]{e}{d}
\arc{d}{u}
\arc{b}{u}



\pspolygon[linecolor=black](14.5,4.5)(14.5,5.5)(15.5,5.5)(15.5,4.5)
\rput[c](15,5){$n$}

\psline[doubleline=true,arrows=->,linecolor=black](7,5.8)(7,7)(5,7)(5,5)(6.2,5)

\rput[l](18,5){Temporal links between non-observed nodes}
\rput[l](18,4){are dangerous!}

%\showgrid
\end{pspicture}
\end{center}

\ul{The unfolded version:}
\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}

\hspace*{-0.6cm}
\begin{pspicture}(6,-2)(36,7)


\Fnode{11}{5}{b1}{$B_1$}
\Fnode{11}{2}{e1}{$E_1$}
\Fnode{7}{5}{a1}{$A_1$}
\Fnode{7}{2}{c1}{$C_1$}
\Fbox{9}{-1}{d1}{$D_1$}
\Fdiam{14}{2}{u1}{$U_1$}

\arc{a1}{c1}
\arc{b1}{c1}
\arc{a1}{e1}
\arc[red]{c1}{d1}
\arc[red]{e1}{d1}
\arc{d1}{u1}
\arc{b1}{u1}


\Fnode{22}{5}{b2}{$B_2$}
\Fnode{22}{2}{e2}{$E_2$}
\Fnode{18}{5}{a2}{$A_2$}
\Fnode{18}{2}{c2}{$C_2$}
\Fbox{20}{-1}{d2}{$D_2$}
\Fdiam{25}{2}{u2}{$U_2$}

\arc{a2}{c2}
\arc{b2}{c2}
\arc{a2}{e2}
\arc[red]{c2}{d2}
\arc[red]{e2}{d2}
\arc{d2}{u2}
\arc{b2}{u2}

\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a1}{a2}
\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a1}{b2}



\Fnode{33}{5}{b3}{$B_3$}
\Fnode{33}{2}{e3}{$E_3$}
\Fnode{29}{5}{a3}{$A_3$}
\Fnode{29}{2}{c3}{$C_3$}
\Fbox{31}{-1}{d3}{$D_3$}
\Fdiam{36}{2}{u3}{$U_3$}

\arc{a3}{c3}
\arc{b3}{c3}
\arc{a3}{e3}
\arc[red]{c3}{d3}
\arc[red]{e3}{d3}
\arc{d3}{u3}
\arc{b3}{u3}

\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a2}{a3}
\ncarc[arrows=->,linecolor=green,arcangleA=20,arcangleB=20]{a2}{b3}



%\showgrid
\end{pspicture}


\end{slide}

}



%% %####################################################################################
%% %# Is this approach always sound?
%% %####################################################################################

%% \begin{slide}{Is this approach always sound?}


%% \begin{itemize}
%% \item Earth quake protection
%% \item Electing a president
%% \item Environmental legislation
%% \item Killing your husband
%% \item Etc. $\ldots$
%% \end{itemize}
%% \ \\
%% \ \\

%% \centerline{
%% \Red{
%% Decision theory provides convincing arguments in favor of the
%% approach!
%% }
%% }


%% \end{slide}



%####################################################################################
%# When are ID's suitable for repeated use
%####################################################################################

\begin{slide}{When are ID's suitable for repeated use}


\begin{itemize}
\item The sequence of decisions $\Red{D_1},\Red{D_2},\ldots ,\Red{D_n}$ is fixed.\newline
\item The \Var{chance variables} in $I_i$ are \ul{always} observed after $\Red{D_i}$
  and before $\Red{D_{i+1}}$.\newline
\item The decision maker remembers the past.\newline
\item The decision problem is symmetric.\newline\newline
The \Red{decision}-\Var{observation} sequence is independent of the actual
\Var{observations} and \Red{decisions}.
\end{itemize}


\end{slide}


%####################################################################################
%# Test decisions
%####################################################################################

\PDForPS{
\overlays{2}{
\begin{slide}{A cause of asymmetry: Test decisions}

\onlySlide*{1}{
Take your temperature before deciding on aspirin.

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(0,0)(19,7)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{a-fe}{A-Fever}
\Fovalnode[0.6cm]{17}{2}{sl}{Sleepy}

\arc{fl}{fe}
\arc{fe}{a-fe}
\arc{a-fe}{sl}


\Fbox[0.7cm]{12}{5}{a}{Aspirin}
\Ftria[0.6cm]{7}{5}{t}{Temp}

\arc{fe}{t}
\arc{a}{a-fe}
\arc[red]{t}{a}

%\showgrid
\end{pspicture}
\end{center}
\ \\
\ \\

\centerline{\Blue{You only observe the test result (Fever) if you decide to
  take your temperature}}


}
\onlySlide*{2}{
But these problems can still be modeled in influence diagrams:

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(0,0)(19,7)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{a-fe}{A-Fever}
\Fovalnode[0.6cm]{17}{2}{sl}{Sleepy}

\arc{fl}{fe}
\arc{fe}{a-fe}
\arc{a-fe}{sl}


\Fbox[0.7cm]{12}{5}{a}{Aspirin}
\Fbox[0.6cm]{2}{5}{tf}{$T_F$}
\Fovalnode{7}{5}{f'}{F'}


\arc{fe}{f'}
\arc{tf}{f'}
\arc[red]{f'}{a}
\arc{a}{a-fe}
\arc[black]{t}{a}

%\showgrid
\end{pspicture}
\end{center}

\begin{center}
\begin{tabular}{cc|cc}
 & & \multicolumn{2}{c}{\Var{Fever}} \\
 & & \St{y} & \St{n} \\ \hline
\multirow{2}{0.2cm}{$\Red{T_F}$} & \St{y} & $(1,0,0)$ & $(0,1,0)$ \\  
& \St{n} & $(0,0,1)$ & $(0,0,1)$ \\
\multicolumn{4}{c}{$P(F'=(\StM{y},\StM{n},\StM{no-t})|\VarM{Fever},\Red{T_F})$}
\end{tabular}
\end{center}


}
\end{slide}
}
}
{

\begin{slide}{Test decisions I}

Take your temperature before deciding on aspirin.

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(0,0)(19,7)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{a-fe}{A-Fever}
\Fovalnode[0.6cm]{17}{2}{sl}{Sleepy}

\arc{fl}{fe}
\arc{fe}{a-fe}
\arc{a-fe}{sl}


\Fbox[0.7cm]{12}{5}{a}{Aspirin}
\Ftria[0.6cm]{7}{5}{t}{Temp}

\arc{fe}{t}
\arc{a}{a-fe}
\arc[red]{t}{a}

%\showgrid
\end{pspicture}
\end{center}
\end{slide}

\begin{slide}{Test decisions II}
Take your temperature before deciding on aspirin.

\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(0,0)(19,7)

\Fovalnode[0.6cm]{2}{2}{fl}{Flue}
\Fovalnode[0.6cm]{7}{2}{fe}{Fever}
\Fovalnode[0.6cm]{12}{2}{a-fe}{A-Fever}
\Fovalnode[0.6cm]{17}{2}{sl}{Sleepy}

\arc{fl}{fe}
\arc{fe}{a-fe}
\arc{a-fe}{sl}


\Fbox[0.7cm]{12}{5}{a}{Aspirin}
\Fbox[0.6cm]{2}{5}{tf}{$T_F$}
\Fovalnode{7}{5}{f'}{F'}


\arc{fe}{f'}
\arc{tf}{f'}
\arc[red]{f'}{a}
\arc{a}{a-fe}
\arc[black]{t}{a}

%\showgrid
\end{pspicture}
\end{center}

\begin{center}
\begin{tabular}{cc|cc}
 & & \multicolumn{2}{c}{\Var{Fever}} \\
 & & \St{y} & \St{n} \\ \hline
\multirow{2}{0.2cm}{$\Red{T_F}$} & \St{y} & $(1,0,0)$ & $(0,1,0)$ \\  
& \St{n} & $(0,0,1)$ & $(0,0,1)$ \\
\multicolumn{4}{c}{$P(F'=(\StM{y},\StM{n},\StM{no-t})|\VarM{Fever},\Red{T_F})$}
\end{tabular}
\end{center}


\end{slide}

}


%####################################################################################
%# Test decisions
%####################################################################################


\begin{slide}{Transformation of test-decisions in general}


\psset{unit=0.4cm}
\newpsobject{showgrid}{psgrid}{subgriddiv=1,griddots=10,gridlabels=6pt}
\begin{center}
\begin{pspicture}(2,0)(29,9)

\Fnode[0.4cm]{7}{2}{a}{A}

\Fnode[0.4cm]{12}{8}{x1}{}
%\Fnode[0.4cm]{16}{5}{x2}{}
\Fnode[0.4cm]{11}{2}{x3}{}
\Fnode[0.4cm]{3}{2}{x4}{}



\Fbox[0.6cm]{12}{5}{d}{$D$}
\Ftria[0.6cm]{7}{5}{t}{$T_A$}

\arc[red]{t}{d}
\arc{a}{t}
\arc{x4}{a}
\arc{a}{x3}
\arc{x1}{d}
\arc{d}{x2}


\Fbox[0.6cm]{20}{5}{ta}{$T_A$}
\Fnode{24}{5}{a'}{$A'$}
\Fbox{28}{5}{d}{$D$}
\Fnode[0.4cm]{20}{2}{a}{A}

\Fnode[0.4cm]{25}{8}{x1}{}
%\Fnode[0.4cm]{29}{5}{x2}{}
\Fnode[0.4cm]{24}{2}{x3}{}
\Fnode[0.4cm]{16}{2}{x4}{}


\arc[red]{ta}{a'}
\arc{a}{a'}
\arc{x4}{a}
\arc{a}{x3}
\arc{x1}{d}
\arc{a'}{d}


%\showgrid
\end{pspicture}
\end{center}



\begin{center}
\begin{tabular}{cc|cccc}
 & & \multicolumn{4}{c}{\Var{A}} \\
 & & $\St{a_1}$ & $\St{a_2}$ & $\cdots$ & $\St{a_n}$  \\ \hline
\multirow{2}{0.2cm}{$\Red{T_A}$} & \St{y} & $(1,0,\ldots ,0)$ &
 $(0,1,0,\ldots ,0)$ &
 $\cdots$ & $(0,\ldots ,0, 1,0)$ \\  
& \St{n} & $(0,\ldots ,1)$ & $(0,\ldots ,1)$ & $\cdots$ & $(0,\ldots
 ,1)$ \\
\multicolumn{6}{c}{$P(F'=(\St{a_1},\ldots ,\St{a_n},\St{no-t})|\VarM{A},\Red{T_A})$}
\end{tabular}
\end{center}

\end{slide}






%% \end{slide}

\end{document}





