\documentclass[a4paper,12pt]{article}
\usepackage{my/page}
\Marginwidth{2.5cm}
\Marginheight{1.5cm}


\begin{document}
\section*{\bf Recognition \hfill June 11,  2007}
\section*{\bf Name: }

\begin{enumerate}
\item 
 Objects from three classes $k \in \{1,2,3\}$ with identical a priori probabilities
 are classified.  Conditional probabilities $P(x|k)$ are normally distributed:
 $N(0,1)$ for $k=1$, $N(0,2)$ for $k=2$ and $N(3,2)$ for $k=3$. 
 The $N(\mu,\sigma)$ notation defines mean value $\mu$ and standard deviation $\sigma$. 
 What is the decision $d$ optimal from the point of view of bayesian risk minimization,
 $d \in \{1,2,3\}$ for observations $x=1$ and $x=0$, 
 
\begin{itemize}
  \item (2 points) 
        if the loss matrix is
        $$W= \left[ \begin{array}{ccc}0 & 1 & 1 \\
                                     1 & 0 & 1 \\
                                     1 & 1 & 0 \end{array} \right]\:.$$
	(minimisation of the number of incorrect decisions, i.e. error probability) and
\item (2 points) if the loss matrix is
    $$W= \left[ \begin{array}{ccc}0 & 2 & 1 \\
                                 2 & 0 & 1 \\
                                 1 & 1 & 0 \end{array} \right]\:.$$

\end{itemize}
What is the probability of incorrect decision for the strategy minimising error
probability  for $x=1$ (1 point)?

\item (5  points) A training set is given, 
    $ T = \{ (\mathbf{x_i};k_i) \}, i=1, \ldots, 5, \mathbf{x_i} \in R^2, k \in \{1,-1\}$, 
    $T=\{(-2,1;-1),(0,0;-1),(0,2;1),(0,-3;-1), (2,2,1) \}$. 
    
    Find a linear classifier by the perceptron algorithm, i.e. 
    find a vector $\mathbf{w} \in R^2$ and offset $b \in R$ such that $
y=\mathbf{w}\mathbf{x}+b$ is positive for sample from class $k=1$ end negative for $k=-1$.

     In particual, what are the vector $\mathbf{w}$ and offset $b$ after ten steps of
     the perceptron algorithm?

\item (5 points) Describe the AdaBoost learning algorithm.

\item (5 points) Reliability of light bulbs is defined by the probability density function 
   $p(t)=\theta e^{-\theta t}, t \in (0,\infty)$, where $p(t)$ is the probability that
   a light bulb blows out at time $t$. What is the maximum likelihood estimate 
   of the reliability parameter $\theta$ if in an experiment with four bulbs
   the following lifetimes have been observed $t_1=32, t_2=158, t_3=256,
   t_4=1130$?

\end{enumerate}
\end{document}


