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

\newcommand{\exercisetopic}{Maximum Likelihood}
\newcommand{\exercisenum}{10}
\newcommand{\exercisedate}{January 12th, 2021}

\input{../../exercisesheader}

\firstpagefooter{Prof. Dr. Jan Benda}{}{jan.benda@uni-tuebingen.de}

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\begin{document}

\input{../../exercisestitle}

\begin{questions}

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\question \qt{Read chapter 9 on ``Maximum likelihood estimation''!}\vspace{-3ex}

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\question \qt{Maximum likelihood of the standard deviation}
Let's compute the likelihood and the log-likelihood for the estimation
of the standard deviation.
\begin{parts}
  \part Draw $n=50$ random numbers from a normal distribution with
  mean $\mu=3$ and standard deviation $\sigma=2$.

  \part Plot the likelihood (computed as the product of probabilities)
  and the log-likelihood (sum of the logarithms of the probabilities)
  as a function of the standard deviation. Compare the position of the
  maxima with the standard deviation that you compute directly from
  the data.

  \part Increase $n$ to 1000. What happens to the likelihood, what
  happens to the log-likelihood? Why?
\end{parts}
\begin{solution}
  \lstinputlisting{mlestd.m}
  \includegraphics[width=1\textwidth]{mlestd}\\

  The more data the smaller the product of the probabilities ($\approx
  p^n$ with $0 \le p < 1$) and the smaller the sum of the logarithms
  of the probabilities ($\approx n\log p$, note that $\log p < 0$).

  The product eventually gets smaller than the precision of the
  floating point numbers support. Therefore for $n=1000$ the products
  become zero. Using the logarithm avoids this numerical problem.
\end{solution}

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\question \qt{Maximum-likelihood estimator of a line through the origin} 
In the lecture we derived the following equation for an
maximum-likelihood estimate of the slope $m$ of a straight line
through the origin fitted to $n$ pairs of data values $(x_i|y_i)$ with
standard deviation $\sigma_i$:
\[ m = \frac{\sum_{i=1}^n \frac{x_i y_i}{\sigma_i^2}}{ \sum_{i=1}^n
  \frac{x_i^2}{\sigma_i^2}} \]
\begin{parts}
  \part \label{mleslopefunc} Write a function that takes two vectors
  $x$ and $y$ containing the data pairs and returns the slope,
  computed according to this equation. For simplicity we assume
  $\sigma_i=\sigma$ for all $1 \le i \le n$. How does this simplify
  the equation for the slope?
  \begin{solution}
    \lstinputlisting{mleslope.m}
  \end{solution}

  \part Write a script that generates data pairs that scatter with a normal
  distribution around a line through the origin with a given
  slope. Use the function from \pref{mleslopefunc} to compute the
  slope from the generated data.  Compare the computed slope with the
  true slope that has been used to generate the data. Plot the data
  together with the line from which the data were generated as well as
  the maximum-likelihood fit.
  \begin{solution}
    \lstinputlisting{mlepropfit.m}
    \includegraphics[width=1\textwidth]{mlepropfit}
  \end{solution}

  \part \label{mleslopecomp} Vary the number of data pairs, the slope,
  as well as the variance of the data points around the true
  line. Under which conditions is the maximum-likelihood estimation of
  the slope closer to the true slope?

  \part To answer \pref{mleslopecomp} more precisely, generate for
  each condition let's say 1000 data sets and plot a histogram of the
  estimated slopes. How does the histogram, its mean and standard
  deviation relate to the true slope?
\end{parts}
\begin{solution}
  \lstinputlisting{mlepropest.m}
  \includegraphics[width=1\textwidth]{mlepropest}\\
  The estimated slopes are centered around the true slope. The
  standard deviation of the estimated slopes gets smaller for larger
  $n$ and less noise in the data.
\end{solution}

\continue
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\question \qt{Maximum-likelihood-estimation of a probability-density function}
Many probability-density functions have parameters that cannot be
computed directly from the data as it is the case for the mean of
normally-distributed data. Such parameter need to be estimated
numerically by means of maximum-likelihood from the data.

Let us demonstrate this approach by means of data that are drawn from a
gamma distribution.
\begin{parts}
  \part Find out which \code{matlab} function computes the
  probability-density function of the gamma distribution.

  \part \label{gammaplot} Use this function to plot the
  probability-density function of the gamma distribution for various
  values of the (positive) ``shape'' parameter. Set the ``scale''
  parameter to one.

  \part Find out which \code{matlab} function generates random numbers
  that are distributed according to a gamma distribution. Generate
  with this function 50 random numbers using one of the values of the
  ``shape'' parameter used in \pref{gammaplot}.

  \part Compute and plot a properly normalized histogram of these
  random numbers.

  \part Find out which \code{matlab} function fits a distribution to a
  vector of random numbers according to the maximum-likelihood method.
  How do you need to use this function in order to fit a gamma
  distribution to the data?

  \part Estimate with this function the parameter of the gamma
  distribution used to generate the data.

  \part Finally, plot the fitted gamma distribution on top of the
  normalized histogram of the data.
\end{parts}
\begin{solution}
  \lstinputlisting{mlepdffit.m}
  \includegraphics[width=1\textwidth]{mlepdffit}
\end{solution}

\end{questions}

\end{document}