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

\usepackage{epsfig}

\setlength{\unitlength}{1cm}
\usepackage{graphicx}

\title{\textcolor{blue}{Computational Intelligence}}
\subtitle{Probabilistic Graphical Models in AI}

\author{Peter Lucas, Marina Velikova and Nivea de Carvalho-Ferreira}
\institution{Institute for Computing and Information Sciences \\
        Radboud University Nijmegen}
\email{peterl@cs.ru.nl,marinav@cs.ru.nl,nivea@cs.ru.nl}
\slideCaption{Lecture 1: Intro}

\newcommand{\idr}{\perp\!\!\!\perp}

\begin{document}

\maketitle

\begin{slide}[R]{Course Information}
\begin{itemize}
\item \textcolor{red}{Lecturers:} Peter Lucas, Marina Velikova and Nivea de Carvalho-Ferreira
\item \textcolor{red}{Structure of course:}
\begin{itemize}
\item Lectures
\item Seminar form: presentations and discussions
\item Practical assignment: develop your own Bayesian network; experiment
      with learning (structure and classifiers)
\end{itemize}
\item \textcolor{red}{Assessment:}
\begin{itemize}
\item Exam: 40\%
\item Practical assignment 1 and 2: 15\% each
\item Seminar: 30\%
\end{itemize}
\item Course information: \textcolor{red}{\textsf{www.cs.ru.nl/$\sim$peterl/teaching/CI/}}
\end{itemize}
\end{slide}

\begin{slide}[R]{Independence Representation in Graphs}
The set of variables $X$ is \textcolor{red}{conditionally
  independent}
of the set $Z$ \emph{given} the set $Y$, notation {\color{red}$X \idr Z \mid Y$},  iff \\
{\color{red}\begin{center} \fbox{$P(X \mid Y, Z) = P(X \mid Y)$}
\end{center}}
\vspace*{0.5cm}
  Meaning:
\begin{quote}
  ``If we know $Y$ then $Z$ does not have any (extra) effect on our
  knowledge concerning $X$ (and thus can be omitted)''
\end{quote}
\vspace*{0.5cm}
\textcolor{red}{Example} \\
If we know that John has fever, then also knowing that he has a high
body temperature has no effect on our knowledge about
flu
\end{slide}

\begin{slide}[R]{Find the Independences}
\epsfxsize=10cm
\epsfbox{sars-unk.ps}
Examples:
\begin{itemize}
\item $\mbox{FLU} \; \idr \; \mbox{VisitToChina} \mid \varnothing$
\item $\mbox{FLU} \; \idr \; \mbox{SARS} \mid \varnothing$
\item {\color{red}$\mbox{FLU} \; \not\!\idr \; \mbox{SARS} \mid \mbox{FEVER}$}, also
{\color{red}$\mbox{FLU} \; \not\!\idr \; \mbox{SARS} \mid \mbox{TEMP}$}
\item $\mbox{SARS} \; \idr \; \mbox{TEMP} \mid \mbox{FEVER}$
\item $\mbox{VisitToChina} \; \idr \; \mbox{DYSPNOEA} \mid \mbox{SARS}$
\end{itemize}
\end{slide}

\begin{slide}[R]{Probabilistic Reasoning}
\begin{itemize}
\item Interested in \textcolor{red}{conditional} probability
  distributions:
\[{\color{red}P(X_W \mid \mathcal{E}) =
    {P}^{\mathcal{E}}(X_W)}\] with $W$ set of vertices,
  for (possibly empty) {\color{red}evidence} $\mathcal{E}$ (instantiated variables) \\[2ex]
  \textcolor{red}{Examples} \\
\[P(\mathrm{FLU} = \textit{yes} \mid \mathrm{TEMP} < 37.5)\]
\[P(\mathrm{FLU} = \textit{yes}, \mathrm{VisitToAsia} = \textit{yes} \mid \mathrm{TEMP} < 37.5)\]
\vspace*{0.1cm}
\item Tendency to focus on conditional probability distributions
      of single variables
\end{itemize}
\end{slide}

\begin{slide}[R]{Probabilistic Reasoning (cont)}
\begin{itemize}
\item Joint probability distribution $P(X)$:
$P(X) = P(X_1,X_2,\ldots,X_n)$
\begin{itemize}
\item {\color{red}marginalisation:}
\begin{eqnarray*}
P(Y) = \sum_{X \backslash Y} P(X) 
         = \sum_{X \backslash Y} \prod_{v\in V} P(X_v \mid X_{\pi(v)})
\end{eqnarray*}
\item {\color{red}conditional probabilities and Bayes' rule:}
\[P(Y,Z  \mid X) = \frac{P(X \mid Y,Z) P(Y,Z)}{P(X)}\]
\end{itemize}
\item Many {\color{red}efficient Bayesian reasoning
  algorithms} exist
\end{itemize}
\end{slide}

\end{document}
