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[projects] updated mutual information and noisy ficurves

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2020-01-22 18:50:47 +01:00
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projects/project_mutualinfo/mutualinfo.tex
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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. It quantifies the
dependence of an output $y$ (e.g. a spike train) on some input $x$
(e.g. a sensory stimulus).
The probability of each of $n$ input values $x = {x_1, x_2, ... x_n}$
is given by the corresponding probabilty distribution $P(x)$. The entropy
\begin{equation}
\label{entropy}
H[x] = - \sum_{x} P(x) \log_2 P(x)
\end{equation}
is a measure for the surprise of getting a specific value of $x$. For
example, if from two possible values '1' and '2', the probability of
getting a '1' is close to one ($P(1) \approx 1$) then the probability
of getting a '2' is close to zero ($P(2) \approx 0$). For this case
the entropy, the surprise level, is almost zero, because both $0 \log
0 = 0$ and $1 \log 1 = 0$. It is not surprising at all that you almost
always get a '1'. The entropy is largest for equally likely outcomes
of $x$. If getting a '1' or a '2' is equally likely then you will be
most surprised by each new number you get, because you can not predict
them.
Mutual information measures information transmitted between an input
and an output. It is computed from the probability distributions of
the input, $P(x)$, the output $P(y)$ and their joint distribution
$P(x,y)$:
\begin{equation}
\label{mi}
I[x:y] = \sum_{x}\sum_{y} P(x,y) \log_2\frac{P(x,y)}{P(x)P(y)}
\end{equation}
where the sums go over all possible values of $x$ and $y$. The mutual
information can be also expressed in terms of entropies. Mutual
information is the entropy of the outputs $y$ reduced by the entropy
of the outputs given the input:
\begin{equation}
\label{mientropy}
I[x:y] = E[y] - E[x|y]
\end{equation}
The following project is meant to explore the concept of mutual
information with the help of a simple example.
\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
@@ -19,50 +62,56 @@
object was reported by the subject.
\begin{parts}
\part Plot the data appropriately.
\part Plot the raw data (no sums or probabilities) appropriately.
\part Compute and plot the probability distributions of presented
and reported objects.
\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.
object and $y$ is the reported object).
\part Use the computed probability distributions to compute the mutual
information \eqref{mi} that the answers provide about the
actually presented object.
\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}
\question What is the maximally achievable mutual information?
\begin{parts}
\part Show this numerically by generating your own datasets which
naturally should yield maximal information. Consider different
distributions of $P(x)$.
\part Compare the maximal mutual information with the corresponding
entropy \eqref{entropy}.
\end{parts}
\question What is the minimum possible mutual information?
This is the mutual information between an output is independent of the
input.
How is the joint distribution $P(x,y)$ related to the marginls
$P(x)$ and $P(y)$ if $x$ and $y$ are independent? What is the value
of the logarithm in eqn.~\eqref{mi} in this case? So what is the
resulting value for the mutual information?
\end{questions}
Hint: 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.
\end{document}