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\chapter{Maximum likelihood estimation}
\label{maximumlikelihoodchapter}
\exercisechapter{Maximum likelihood estimation}

The core task of statistics is to infer from measured data some
parameters describing the data. These parameters can be simply a mean,
a standard deviation, or any other parameter needed to describe the
distribution the data are originating from, a correlation coefficient,
or some parameters of a function describing a particular dependence
between the data. The brain faces exactly the same problem. Given the
activity pattern of some neurons (the data) it needs to infer some
aspects (parameters) of the environment and the internal state of the
body in order to generate some useful behavior. One possible approach
to estimate parameters from data are \enterm[maximum likelihood
estimator]{maximum likelihood estimators} (\enterm[mle|see{maximum
  likelihood estimator}]{mle},
\determ{Maximum-Likelihood-Sch\"atzer}).  They choose the parameters
such that they maximize the likelihood of the specific data values to
originate from a specific distribution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Maximum likelihood}

Let $p(x|\theta)$ (to be read as ``probability(density) of $x$ given
$\theta$.'') the probability (density) distribution of data value $x$
given parameter values $\theta$. This could be the normal distribution
\begin{equation}
  \label{normpdfmean}
  p(x|\mu, \sigma) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}
\end{equation}
defined by the mean $\mu$ and the standard deviation $\sigma$ as
parameters $\theta$.  If the $n$ observations $x_1, x_2, \ldots, x_n$
are independent of each other and originate from the same probability
density distribution (they are \enterm[i.i.d.|see{independent and
  identically distributed}]{i.i.d.}, \enterm{independent and
  identically distributed}), then the conditional probability
$p(x_1,x_2, \ldots, x_n|\theta)$ of observing the particular data
values $x_1, x_2, \ldots, x_n$ given some specific parameter values
$\theta$ of the probability density is given by the product of the
probability densities of each data value:
\begin{equation}
  \label{probdata}
  p(x_1,x_2, \ldots, x_n|\theta) = p(x_1|\theta) \cdot p(x_2|\theta)
  \ldots, p(x_n|\theta) = \prod_{i=1}^n p(x_i|\theta) \; .
\end{equation}
Vice versa, the \entermde{Likelihood}{likelihood} of the parameters $\theta$
given the observed data $x_1, x_2, \ldots, x_n$ is
\begin{equation}
  \label{likelihood}
  {\cal L}(\theta|x_1,x_2, \ldots, x_n) = p(x_1,x_2, \ldots, x_n|\theta) \; .
\end{equation}
Note, that the likelihood ${\cal L}$ is not a probability in the
classic sense since it does not integrate to unity ($\int {\cal
  L}(\theta|x_1,x_2, \ldots, x_n) \, d\theta \ne 1$). For given
observations $x_1, x_2, \ldots, x_n$, the likelihood
\eqref{likelihood} is a function of the parameters $\theta$. This
function has a global maximum for some specific parameter values. At
this maximum the probability \eqref{probdata} to observe the measured
data values is the largest.

Maximum likelihood estimators just find the parameter values
\begin{equation}
  \theta_{mle} = \text{argmax}_{\theta} {\cal L}(\theta|x_1,x_2, \ldots, x_n)
\end{equation}
that maximize the likelihood \eqref{likelihood}.
$\text{argmax}_xf(x)$ is the value of the argument $x$ for which the
function $f(x)$ assumes its global maximum. Thus, we search for the
parameter values $\theta$ at which the likelihood ${\cal L}(\theta)$
reaches its maximum. For these parameter values the measured data most
likely originated from the corresponding distribution.

The position of a function's maximum does not change when the values
of the function are transformed by a strictly monotonously rising
function such as the logarithm. For numerical reasons and reasons that
we discuss below, we instead search for the maximum of the logarithm
of the likelihood
(\entermde[likelihood!log-]{Likelihood!Log-}{log-likelihood})
\begin{eqnarray}
  \theta_{mle} &amp; = &amp; \text{argmax}_{\theta}\; {\cal L}(\theta|x_1,x_2, \ldots, x_n) \nonumber \\
              &amp; = &amp; \text{argmax}_{\theta}\; \log {\cal L}(\theta|x_1,x_2, \ldots, x_n) \nonumber \\
              &amp; = &amp; \text{argmax}_{\theta}\; \log \prod_{i=1}^n p(x_i|\theta) \nonumber \\
              &amp; = &amp; \text{argmax}_{\theta}\; \sum_{i=1}^n \log p(x_i|\theta) \label{loglikelihood}
\end{eqnarray}
which is the sum of the logarithms of the probabilites of each
observation. Let's illustrate the concept of maximum likelihood
estimation on the arithmetic mean.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Arithmetic mean}
Suppose that the measurements $x_1, x_2, \ldots, x_n$ originate from a
normal distribution \eqref{normpdfmean} and we do not know the
population mean $\mu$ of the normal distribution
(\figrefb{mlemeanfig}). In this setting $\mu$ is the only parameter
$\theta$.  Which value of $\mu$ maximizes the likelihood of the data?

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{mlemean}
  \titlecaption{\label{mlemeanfig} Maximum likelihood estimation of
    the mean.}{Top: The measured data (blue dots) together with three
    normal distributions differing in their means (arrows) from which
    the data could have originated from.  Bottom left: the likelihood
    as a function of the parameter $\mu$. For the data it is maximal
    at a value of $\mu = 2$. Bottom right: the log-likelihood. Taking
    the logarithm does not change the position of the maximum.}
\end{figure}

With the normal distribution \eqref{normpdfmean} and applying
logarithmic identities, the log-likelihood \eqref{loglikelihood} reads
\begin{eqnarray}
  \log {\cal L}(\mu|x_1,x_2, \ldots, x_n)
  &amp; = &amp; \sum_{i=1}^n \log \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x_i-\mu)^2}{2\sigma^2}} \nonumber \\
  &amp; = &amp; \sum_{i=1}^n - \log \sqrt{2\pi \sigma^2} -\frac{(x_i-\mu)^2}{2\sigma^2} \; .
\end{eqnarray}
Since the logarithm is the inverse function of the exponential
($\log(e^x)=x$), taking the logarithm removes the exponential from the
normal distribution. This is the second reason why it is useful to
maximize the log-likelihood. To calculate the maximum of the
log-likelihood, we need to take the derivative with respect to $\mu$
and set it to zero:
\begin{eqnarray}
  \frac{\text{d}}{\text{d}\mu} \log {\cal L}(\mu|x_1,x_2, \ldots, x_n) &amp; = &amp; \sum_{i=1}^n - \frac{\text{d}}{\text{d}\mu} \log \sqrt{2\pi \sigma^2} - \frac{\text{d}}{\text{d}\mu} \frac{(x_i-\mu)^2}{2\sigma^2} \;\; = \;\;  0  \nonumber \\
  \Leftrightarrow \quad \sum_{i=1}^n - \frac{2(x_i-\mu)}{2\sigma^2} &amp; = &amp; 0  \nonumber \\
  \Leftrightarrow \quad \sum_{i=1}^n x_i - \sum_{i=1}^n \mu &amp; = &amp; 0  \nonumber \\
  \Leftrightarrow \quad n \mu &amp; = &amp; \sum_{i=1}^n x_i  \nonumber \\
  \Leftrightarrow \quad \mu &amp; = &amp; \frac{1}{n} \sum_{i=1}^n x_i \;\; = \;\; \bar x \label{arithmeticmean}
\end{eqnarray}
Thus, the maximum likelihood estimator of the population mean of
normally distributed data is the arithmetic mean. That is, the
arithmetic mean maximizes the likelihood that the data originate from
a normal distribution centered at the arithmetic mean
(\figref{mlemeanfig}). Equivalently, the standard deviation computed
from the data, maximizes the likelihood that the data were generated
from a normal distribution with this standard deviation.

\begin{exercise}{mlemean.m}{mlemean.out}
  Draw $n=50$ random numbers from a normal distribution with a mean of
  $\ne 0$ and a standard deviation of $\ne 1$.

  Plot the likelihood (the product of the probabilities) and the
  log-likelihood (given by the sum of the logarithms of the
  probabilities) for the mean as parameter. Compare the position of
  the maxima with the mean calculated from the data.
\end{exercise}

Comparing the values of the likelihood with the ones of the
log-likelihood shown in \figref{mlemeanfig}, shows the numerical
reason for taking the logarithm of the likelihood. The likelihood
values can get very small, because we multiply many, potentially small
probability densities with each other. The likelihood quickly gets
smaller than the samlles number a floating point number of a computer
can represent. Try it by increasing the number of data values in the
exercise. Taking the logarithm avoids this problem. The log-likelihood
assumes well behaving numbers that can be handled well by the
computer.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fitting probability distributions}
Consider normally distributed data with unknown mean and standard
deviation.  From the considerations above we just have seen that a
Gaussian distribution with mean at the arithmetic mean and standard
deviation equal to the standard deviation computed from the data is
the Gaussian that fits the data best in a maximum likelihood sense,
i.e. the likelihood that the data were generated from this
distribution is the largest. Fitting a Gaussian distribution to data
is very simple: just compute the two parameter of the Gaussian
distribution $\mu$ and $\sigma$ as the arithmetic mean and a standard
deviation, respectively, directly from the data.

For non-Gaussian distributions, for example a
\entermde[distribution!Gamma-]{Verteilung!Gamma-}{Gamma-distribution}
\begin{equation}
  \label{gammapdf}
  p(x|\alpha,\beta) \sim x^{\alpha-1}e^{-\beta x}
\end{equation}
with a shape parameter $\alpha$ and a rate parameter $\beta$
(\figrefb{mlepdffig}), in general no such simple analytical
expressions for estimating the parameters directly from the data do
not exist. How do we fit such a distribution to the data?  That is,
how should we compute the values of the parameters of the
distribution, given the data?

A first guess could be to fit the probability density function by a
\enterm{least squares} fit to a normalized histogram of the measured data in
the same way as we fit a function to some data. For several reasons
this is, however, not the method of choice: (i) Probability densities
can only be positive which leads, in particular for small values, to
asymmetric distributions of the estimated histogram around the true
density. (ii) The values of a histogram estimating the density are not
independent because the integral over a density is unity. The two
basic assumptions of normally distributed and independent samples,
which are a prerequisite for making the minimization of the squared
difference to a maximum likelihood estimation (see next section), are
violated. (iii) The estimation of the probability density by means of
a histogram strongly depends on the chosen bin size.

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{mlepdf}
  \titlecaption{\label{mlepdffig} Maximum likelihood estimation of a
    probability density.}{Left: the 100 data points drawn from a 2nd
    order Gamma-distribution. The maximum likelihood estimation of the
    probability density function is shown in orange, the true pdf is
    shown in red. Right: normalized histogram of the data together
    with the true probability density (red) and the probability
    density function obtained by a least squares fit to the
    histogram.}
\end{figure}

Instead we should stay with maximum-likelihood estimation.  Exactly in
the same way as we estimated the mean value of a Gaussian distribution
above, we can numerically fit the parameter of any type of
distribution directly from the data by means of maximizing the
likelihood.  We simply search for the parameter values of the desired
probability density function that maximize the log-likelihood. In
general this is a non-linear optimization problem that is solved with
numerical methods such as the gradient descent \matlabfun{mle()}.

\begin{exercise}{mlegammafit.m}{mlegammafit.out}
  Generate a sample of gamma-distributed random numbers and apply the
  maximum likelihood method to estimate the parameters of the gamma
  function from the data.
\end{exercise}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Curve fitting}

When fitting a function of the form $f(x;\theta)$ to data pairs
$(x_i|y_i)$ one tries to adapt the parameters $\theta$ such that the
function best describes the data. In
chapter~\ref{gradientdescentchapter} we simply assumed that ``best''
means minimizing the squared distance between the data and the
function.  With maximum likelihood we search for those parameter
values $\theta$ that maximize the likelihood of the data to be drawn
from the corresponding function.

If we assume that the $y_i$ values are normally distributed around the
function values $f(x_i;\theta)$ with a standard deviation $\sigma_i$,
the log-likelihood is
\begin{eqnarray}
  \log {\cal L}(\theta|(x_1,y_1,\sigma_1), \ldots, (x_n,y_n,\sigma_n))
  &amp; = &amp; \sum_{i=1}^n \log \frac{1}{\sqrt{2\pi \sigma_i^2}}e^{-\frac{(y_i-f(x_i;\theta))^2}{2\sigma_i^2}}  \nonumber \\
  &amp; = &amp; \sum_{i=1}^n - \log \sqrt{2\pi \sigma_i^2} -\frac{(y_i-f(x_i;\theta))^2}{2\sigma_i^2}
\end{eqnarray}
The only difference to the previous example of the arithmetic mean is
that the means $\mu$ in the equations above are given by the function
values $f(x_i;\theta)$. The parameters $\theta$ should maximize the
log-likelihood. The first term in the sum is independent of $\theta$
and can be ignored when computing the the maximum. From the second
term we pull out the constant factor $-\frac{1}{2}$:
\begin{eqnarray}
  &amp; = &amp; - \frac{1}{2} \sum_{i=1}^n \left( \frac{y_i-f(x_i;\theta)}{\sigma_i} \right)^2
\end{eqnarray}
We can further simplify by inverting the sign and then search for the
minimum. Also the factor $\frac{1}{2}$ can be ignored since it does
not affect the position of the minimum:
\begin{equation}
  \label{chisqmin}
  \theta_{mle} = \text{argmin}_{\theta} \; \sum_{i=1}^n \left( \frac{y_i-f(x_i;\theta)}{\sigma_i} \right)^2 \;\; = \;\; \text{argmin}_{\theta} \; \chi^2
\end{equation}
The sum of the squared differences between the $y$-data values and the
function values, normalized by the standard deviation of the data
around the function, is called $\chi^2$ (chi squared). The parameter
$\theta$ which minimizes the squared differences is thus the one that
maximizes the likelihood of the data to actually originate from the
given function. 

Whether minimizing $\chi^2$ or the \enterm{mean squared error}
\eqref{meansquarederror} introduced in
chapter~\ref{gradientdescentchapter} does not matter. The latter is
the mean and $\chi^2$ is the sum of the squared differences. They
simply differ by the constant factor $n$, the number of data pairs,
which does not affect the position of the minimum. $\chi^2$ is more
general in that it allows for different standard deviations for each
data pair. If they are all the same ($\sigma_i = \sigma$), the common
standard deviation can be pulled out of the sum and also does not
influence the position of the minimum. Both \enterm{least squares} and
minimizing $\chi^2$ are maximum likelihood estimators of the
parameters $\theta$ of a function.

From the mathematical considerations above we can see that the
minimization of the squared difference is a maximum-likelihood
estimation only if the data are normally distributed around the
function. In case of other distributions, the log-likelihood
\eqref{loglikelihood} needs to be adapted accordingly.

\begin{figure}[t]
  \includegraphics[width=1\textwidth]{mlepropline}
  \titlecaption{\label{mleproplinefig} Maximum likelihood estimation
    of the slope of a line through the origin.}{The data (blue and
    left histogram) originate from a straight line $y=mx$ trough the origin
    (red). The maximum-likelihood estimation of the slope $m$ of the
    regression line (orange), \eqnref{mleslope}, is close to the true
    one. The residuals, the data minus the estimated line (right), reveal
    the normal distribution of the data around the line (right histogram).}
\end{figure}

Let's go on and calculate the minimum \eqref{chisqmin} of $\chi^2$
analytically for a simple function.


\subsection{Straight line trough the origin}
The function of a straight line with slope $m$ through the origin
is
\[ f(x) = m x \; . \]
With this specific function, $\chi^2$ reads
\[ \chi^2 = \sum_{i=1}^n \left( \frac{y_i-m x_i}{\sigma_i} \right)^2
\; . \] To calculate the minimum we take the first derivative with
respect to $m$ and equate it to zero:
\begin{eqnarray}
  \frac{\text{d}}{\text{d}m}\chi^2 &amp; = &amp; \sum_{i=1}^n \frac{\text{d}}{\text{d}m} \left( \frac{y_i-m x_i}{\sigma_i} \right)^2 \nonumber \\
  &amp; = &amp; -2 \sum_{i=1}^n  \frac{x_i}{\sigma_i} \left( \frac{y_i-m x_i}{\sigma_i} \right) \nonumber \\
  &amp; = &amp; -2 \sum_{i=1}^n \left( \frac{x_i y_i}{\sigma_i^2} - m \frac{x_i^2}{\sigma_i^2} \right) \;\; = \;\; 0 \nonumber \\
\Leftrightarrow \quad  m \sum_{i=1}^n \frac{x_i^2}{\sigma_i^2} &amp; = &amp; \sum_{i=1}^n \frac{x_i y_i}{\sigma_i^2} \nonumber \\
\Leftrightarrow \quad  m &amp; = &amp; \frac{\sum_{i=1}^n \frac{x_i y_i}{\sigma_i^2}}{ \sum_{i=1}^n \frac{x_i^2}{\sigma_i^2}} \label{mleslope}
\end{eqnarray}
This is an analytical expression for the estimation of the slope $m$
of the regression line (\figref{mleproplinefig}). We do not need to
apply numerical methods like the gradient descent to find the slope
that minimizes the squared differences. Instead, we can compute the
slope directly from the data by means of \eqnref{mleslope}, very much
like we calculate the mean of some data by means of the arithmetic
mean \eqref{arithmeticmean}.

\subsection{Linear and non-linear fits}
A gradient descent, as we have done in the previous chapter, is not
necessary for fitting the slope of a straight line, because the slope
can be directly computed via \eqnref{mleslope}. More generally, this
is the case also for fitting the coefficients of linearly combined
basis functions as for example the slope $m$ and the y-intercept $b$
of a straight line
\[ y = m \cdot x +b \]
or the coefficients $a_k$ of a polynomial
\[ y = \sum_{k=0}^N a_k x^k = a_o + a_1x + a_2x^2 + a_3x^4 + \ldots \]
\matlabfun{polyfit()}.

Parameters that are non-linearly combined can not be calculated
analytically. Consider, for example, the factor $c$ and the rate
$\lambda$ of the exponential decay
\[ y = c \cdot e^{\lambda x} \quad , \quad c, \lambda \in \reZ \; . \]
Such cases require numerical solutions for the optimization of the
cost function, e.g. the gradient descent \matlabfun{lsqcurvefit()}.

\subsection{Relation between slope and correlation coefficient}
Let us have a closer look on \eqnref{mleslope} for the slope of a line
through the origin. If the standard deviation of the data $\sigma_i$
is the same for each data point, i.e. $\sigma_i = \sigma_j \; \forall
\; i, j$, the standard deviation drops out and \eqnref{mleslope}
simplifies to
\begin{equation}
  \label{whitemleslope}
  m = \frac{\sum_{i=1}^n x_i y_i}{\sum_{i=1}^n x_i^2}
\end{equation}
To see what the nominator and the denominator of this expression
describe, we need to subtract from the data their mean value. We make
the data mean free, i.e. $x \mapsto x - \bar x$ and $y \mapsto y -
\bar y$ . For mean-free data the variance
\begin{equation}
  \sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 = \frac{1}{n} \sum_{i=1}^n x_i^2
\end{equation}
is given by the mean squared data.  In the same way, the covariance
between $x$ and $y$ simplifies to
\begin{equation}
  \text{cov}(x, y) = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)(y_i -
  \bar y) =\frac{1}{n} \sum_{i=1}^n x_i y_i \; ,
\end{equation}
the averaged product between pairs of $x$ and $y$ values. Expanding
the fraction in \eqnref{whitemleslope} by $\frac{1}{n}$ we get
\begin{equation}
  \label{meanfreeslope}
  m = \frac{\frac{1}{n}\sum_{i=1}^n x_i y_i}{\frac{1}{n}\sum_{i=1}^n x_i^2}
    = \frac{\text{cov}(x, y)}{\sigma_x^2}
\end{equation}

Recall that the correlation coefficient $r_{x,y}$ is the covariance
normalized by the product of the standard deviations of $x$ and $y$:
\begin{equation}
  \label{meanfreecorrcoef}
  r_{x,y} = \frac{\text{cov}(x, y)}{\sigma_x\sigma_y}
\end{equation}
If furthermore the standard deviations of $x$ and $y$ are the same,
i.e. $\sigma_x = \sigma_y$, the slope of a line trough the origin is
identical to the correlation coefficient.

This relation between slope and correlation coefficient in particular
holds for standardized data that have been made mean free and have
been normalized by their standard deviation, i.e. $x \mapsto (x - \bar
x)/\sigma_x$ and $y \mapsto (y - \bar x)/\sigma_y$. The resulting
numbers are called \entermde[z-value]{z-Wert}{$z$-values} or
\enterm[z-score]{$z$-scores}. Their mean equals zero and their
standard deviation one. $z$-scores are often used to make quantities
that differ in their units comparable. For standardized data the
denominators for both the slope \eqref{meanfreeslope} and the
correlation coefficient \eqref{meanfreecorrcoef} equal one.  For
standardized data, both the slope of the regression line and the
corelation coefficient reduce to the covariance between the $x$ and
$y$ data:
\begin{equation}
  m = \frac{1}{n} \sum_{i=1}^n x_i y_i = \text{cov}(x,y) = r_{x,y}
\end{equation}
For standardized data the slope of the regression line is the same as the
correlation coefficient!


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Neural coding}
Maximum likelihood estimators are not only a central concept for data
analysis. Neural systems face the very same problem. They also need to
estimate parameters of the internal and external environment based on
the activity of neurons.

\begin{figure}[tp]
  \includegraphics[width=1\textwidth]{mlecoding}
  \titlecaption{\label{mlecodingfig} Maximum likelihood estimation of
    a stimulus parameter from neuronal activity.}{Top: Tuning curve
    $r_i(\phi;c,\phi_i)$ of a specific neuron $i$ as a function of the
    orientation $\phi$ of a stimulus, a dark bar in front of a white
    background. The preferred stimulus $\phi_i$ of that neuron, the
    one that evokes the strongest firing rate response, is a bar with
    vertical orientation (arrow, $\phi_i=90$\,\degree). The width of
    the red area indicates the variability of the neuronal activity
    $\sigma_i$ around the tuning curve. Center: In a population of
    neurons, each neuron may have a different tuning curve (colors). A
    specific stimulus activates the individual neurons of the
    population in a specific way (dots). Bottom: The log-likelihood of
    the activity pattern has a maximum close to the real stimulus
    orientation.}
\end{figure}

In sensory systems certain aspects of the environment are encoded in
the neuronal activity of populations of neurons. One example of such a
population code is the tuning of neurons in the primary visual cortex
(V1) to the orientation of a bar in the visual stimulus. Different
neurons respond best to different bar orientations. Traditionally,
such a tuning is measured by analyzing the neuronal response strength
(e.g. the firing rate) as a function of the orientation of a black bar
and is illustrated and summarized with the so called
\enterm{tuning-curve} (\determ{Abstimmkurve},
figure~\ref{mlecodingfig}, top).

The brain, however, is confronted with the inverse problem: given a
certain activity pattern in the neuronal population, what is the
stimulus? In our example, what is the orientation of the bar? In the
sense of maximum likelihood, a possible answer to this question would
be: the stimulus for which the particular activity pattern is most
likely given the tuning of the neurons and the noise (standard
deviation) of the responses.

Let's stay with the example of the orientation tuning in V1. The
tuning of the firing rate $r_i(\phi)$ of neuron $i$ to the preferred
bar orientation $\phi_i$ can be well described using a van-Mises
function (the Gaussian function on a cyclic x-axis)
(\figref{mlecodingfig}):
\begin{equation}
  \label{bartuningcurve}
  r_i(\phi; c, \phi_i) = c \cdot e^{\cos(2(\phi-\phi_i))} \quad , \quad c &gt; 0
\end{equation} 
Note the factor two in the cosine, because the response of the neuron
is the same for a bar turned by 180\,\degree.

If we approximate the neuronal activity by a normal distribution
around the tuning curve with a standard deviation $\sigma_i$, then the
probability $p_i(r|\phi)$ of the $i$-th neuron having the observed
activity $r$, given a certain orientation $\phi$ of a bar is
\begin{equation}
  p_i(r|\phi) = \frac{1}{\sqrt{2\pi\sigma_i^2}} e^{-\frac{1}{2}\left(\frac{r-r_i(\phi; c, \phi_i)}{\sigma_i}\right)^2} \; .
\end{equation}
The log-likelihood of the bar orientation $\phi$ given the
activity pattern in the population $r_1$, $r_2$, ... $r_n$ is thus
\begin{equation}
  {\cal L}(\phi|r_1, r_2, \ldots, r_n) = \sum_{i=1}^n \log p_i(r_i|\phi)
\end{equation}
The angle $\phi_{mle}$ that maximizes this likelihood is an estimate
of the true orientation of the bar (\figref{mlecodingfig}).

The noisiness of the neuron's responses as quantified by $\sigma_i$
usually is a function of the neuron's mean firing rate $r_i$,
\eqnref{bartuningcurve}: $\sigma_i = \sigma_i(r_i)$. This dependence
has a major impact of the maximum likelihood estimation. Usually, the
stronger the response of a neuron, the higher its firing rate, the
lower the noise. In this case, strong responses will have a stronger
influence on the position of the maximum of the log-likelihood.

Whether neural systems really implement maximum likelihood estimators
is another question. There are many ways how a stimulus property can
be read out from the activity of a population of neurons. The simplest
one being a ``winner takes all'' rule. The preferred stimulus
parameter of the neuron with the strongest response is the
estimate. Another possibility is to compute a population vector. The
estimated stimulus parameter is the sum of the preferred stimulus
parameters of all neurons in the population weighted by the activity
of the neurons. In case of angular stimulus parameters, like the
orientation of the bar in our example, a vector pointing in the
direction of the angle is used instead of the angle to incorporate the
cyclic nature of the parameter. 

Using maximum likelihood estimators for analyzing neural population
activity gives us an upper bound of how well a stimulus parameter is
encoded in the activity of the neurons. The brain would not be able to
do better.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\printsolutions
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