\documentclass{article}
\input {../exercise-preamble}
\begin{document}
\psetnum{1}
\date{2005/11/17}

\begin{pset}

  We begin by considering the first four data points. After
  implementing the E-M algorithm\footnote{Python source code for the
    implementation is available at
    \url{http://www.ambulatoryclam.net/svn/classes/6.825/proj3exercises/ex1}},
  we run it for four iterations on the data points. The results are
  shown below.
  
  \begin{center}
    \begin{tabular}{|c||c|c|c|c||c|c|c|c|c|c||c|}
      \hline
      t & $p_{11}$ & $p_{12}$ & $p_{13}$ & $p_{14}$ &
      $p_1$ & $\mu_1$& $\sigma_1^2$ &
      $p_2$ & $\mu_2$& $\sigma_2^2$ &
      $\log p(\mathfrak{X}\;|\;\Theta_t)$ \\
      \hline
      \input{table}
      \hline
    \end{tabular}    
  \end{center}

  \paragraph{Extra credit}
  Using all the data, we test how long the E-M procedure requires to
  converge. We define the procedure to have converged when $p_i$,
  $mu_i$, and $\sigma_i^2$ do not change by more than $\delta$ after
  an E-M step. For $\delta = 0.01$, 12 iterations are required; for
  $\delta = 0.001$, 66 iterations are required; for $\delta = 0.0001$,
  543 iterations are required; and for $\delta = 0.00001$, 813
  iterations are required. The results reached are shown below:

  \begin{center}
    \begin{tabular}{|c||c|c|c|c|c|c||c|}
      \hline
      t &
      $p_1$ & $\mu_1$& $\sigma_1^2$ &
      $p_2$ & $\mu_2$& $\sigma_2^2$ &
      $\log p(\mathfrak{X}\;|\;\Theta_t)$ \\
      \hline
12 &
 0.2287 &
 -0.1549 &
 0.3657 &
 0.7713 &
 2.5329 &
 0.6593 &
-48.2469 \\
66 &
 0.2552 &
 -0.0287 &
 0.4846 &
 0.7448 &
 2.5853 &
 0.5963 &
-48.2223 \\
543 &
 0.3000 &
 0.1714 &
 0.6681 &
 0.7000 &
 2.6668 &
 0.5120 &
-48.2175 \\
813 &
 0.3026 &
 0.1820 &
 0.6778 &
 0.6974 &
 2.6713 &
 0.5076 &
-48.2174 \\
      \hline
    \end{tabular}    
  \end{center}

  
\end{pset}

\end{document}
