\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?

    Show this numerically by generating your own datasets which
    naturally should yield maximal information. Consider different
    distributions of $P(x)$.

    Here you may encounter a problem when computing the mutual
    information whenever $P(x,y)$ equals zero. For treating this
    special case think about (plot it) what the limit of $x \log x$ is
    for $x$ approaching zero. Use this information to fix the
    computation of the mutual information.

    \part Use a permutation test to compute the $95\%$ confidence
    interval for the mutual information estimate in the dataset from
    {\tt decisions.mat}. Does the measured mutual information indicate
    signifikant information transmission?

  \end{parts}

\end{questions}





\end{document}