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\section*{\bf RPZ \hfill 15. January 2013}
\section*{\bf Name: }

\begin{enumerate}
\item (6 points) In $N$ throws of a coin,  the tail was observed
    $k$-times. What is the maximum likelihood estimate of the probability
    of observing a tail in a throw (1 point)?
    Prove it (2 points). What is the Bayesian estimate of the
  probability of observing a tail in a throw
  (1 points)? Outline the proof (2 points).

\item (6 points)
  Formulate the Neyman-Pearson problem (2 points).\\
  What is the optimal strategy for the Neyman-Pearson problem where the objects
  are from two classes $k \in \{A,B\}$. 
  A real number $x \in (-1,1)$ is observed.
  The conditional probability densities $p(x|k)$ have the following
  distributions: 
    $$p(x|A)=\frac{1}{2}, p(x|B)=\frac{3}{2}x^2.$$
  The maximum allowed probability of overlooked danger is 0.1 (10\%); 
  class B is considered dangerous (3 points).
  What is the solution if class A is considered dangerous (1 points).

\item (6 points) A training set is given:
     $ T = \{ (\mathbf{x_i};k_i) \}$, $i=1, \ldots, 5,$ 
    $\mathbf{x_i} \in R^2$,
    $k \in \{1,-1\}$,\\ 
    $T=\{(-2,1;-1),(1,0;1),(0,2;1),(0,-1;1), (2,2,-1) \}$.

The Adaboost algorithm is employed for finding a linear combination of
weak classifiers:\\
$H(\mathbf{x})=\sum \alpha_t h_t(\mathbf{x})$. 
The following weak classifiers are available:
   $h_{1\theta}(\mathbf{x}) =\dd(x_1 >\theta),  
    h_{2\theta}(\mathbf{x}) =\dd(x_1 \le \theta)$ a 
   $h_3(\mathbf{x}) =\dd(x^2_2 + x^2_1 < 4.5),  
    h_4(\mathbf{x}) =\dd(x^2_2 + x^2_1 \ge 4.5)$,
where $\dd()$ equals 1 if the condition in the parentheses is satisfied,
-1 otherwise; $\theta$ is a real number.

Find $\alpha_t$ and $H()$.

\item (4 points). 
 What class of recognition problems are best solved by neural nets (as
 opposed to SVM, AdaBoost, nearest neighbour)? What is the reason?
    
\end{enumerate}
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