59 lines
1.9 KiB
TeX
59 lines
1.9 KiB
TeX
\documentclass[a4paper,12pt,pdftex]{exam}
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\newcommand{\ptitle}{Mutual information}
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\input{../header.tex}
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\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
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{email: jan.benda@uni-tuebingen.de}
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\begin{document}
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\input{../instructions.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{questions}
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\question A subject was presented two possible objects for a very
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brief time ($50$\,ms). The task of the subject was to report which of
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the two objects was shown. In {\tt decisions.mat} you find an array
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that stores which object was presented in each trial and which
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object was reported by the subject.
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\begin{parts}
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\part Plot the data appropriately.
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\part Compute a 2-d histogram that shows how often different
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combinations of reported and presented came up.
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\part Normalize the histogram such that it sums to one (i.e. make
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it a probability distribution $P(x,y)$ where $x$ is the presented
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object and $y$ is the reported object). Compute the probability
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distributions $P(x)$ and $P(y)$ in the same way.
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\part Use that probability distribution to compute the mutual
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information
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\[ I[x:y] = \sum_{x\in\{1,2\}}\sum_{y\in\{1,2\}} P(x,y)
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\log_2\frac{P(x,y)}{P(x)P(y)}\]
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that the answers provide about the actually presented object.
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The mutual information is a measure from information theory that is
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used in neuroscience to quantify, for example, how much information
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a spike train carries about a sensory stimulus.
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\part What is the maximally achievable mutual information (try to
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find out by generating your own dataset which naturally should
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yield maximal information)?
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\part Use bootstrapping (permutation test) to compute the $95\%$
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confidence interval for the mutual information estimate in the
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dataset from {\tt decisions.mat}.
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\end{parts}
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\end{questions}
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\end{document}
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