\documentclass[a4paper,12pt,pdftex]{exam}

\newcommand{\ptitle}{Mutual information}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
{email: jan.benda@uni-tuebingen.de}

\begin{document}

\input{../instructions.tex}


%%%%%%%%%%%%%% 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. 

    The mutual information is a measure from information theory that is
    used in neuroscience to quantify, for example, how much information
    a spike train carries about a sensory stimulus.
    \part What is the maximally achievable mutual information (try to
    find out by generating your own dataset which naturally should
    yield maximal information)?
    \part Use bootstrapping to compute the $95\%$ confidence interval
    for the mutual information estimate in that dataset. 
  \end{parts}

\end{questions}





\end{document}