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\firstpageheader{Scientific Computing}{Project Assignment}{11/05/2014
  -- 11/06/2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\begin{document}
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  \input{../disclaimer.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{questions}
  \question A subject was presented two possible objects for a very
  brief time ($50$ms). The task of the subject was to report which of
  the two objects was shown. In {\tt decisions.mat} you find an array
  that stores which object was presented in each trial and which
  object was reported by the subject. 
  
  \begin{parts}
    \part Plot the data appropriately. 
    \part Compute a 2-d histogram that shows how often different
    combinations of reported and presented came up. 
    \part Normalize the histogram such that it sums to one (i.e. make
    it a probability distribution $P(x,y)$ where $x$ is the presented
    object and $y$ is the reported object). Compute the probability
    distributions $P(x)$ and $P(y)$ in the same way.
    \part Use that probability distribution to compute the mutual
    information $$I[x:y] = \sum_{x\in\{1,2\}}\sum_{y\in\{1,2\}} P(x,y)
    \log_2\frac{P(x,y)}{P(x)P(y)}$$ that the answers provide about the
    actually presented object. 
    \part What is the maximally achievable mutual information (try to
    find out by generating your own dataset; the situation in which
    the information is maximal is pretty straightforward)?
    \part Use bootstrapping to compute the $95\%$ confidence interval
    for the mutual information estimate in that dataset. 
  \end{parts}

\end{questions}





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