[likelihood] fixed plots for exercise

likelihood/exercises/exercises01.tex

@@ -107,7 +107,15 @@ of the standard deviation.
 \end{parts}
 \begin{solution}
   \lstinputlisting{mlestd.m}
-  \includegraphics[width=1\textwidth]{mlestd}
+  \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
+  becomes zero. Using the logarithm avoids this numerical problem.
 \end{solution}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -151,7 +159,10 @@ standard deviation $\sigma_i$:
 \end{parts}
 \begin{solution}
   \lstinputlisting{mlepropest.m}
-  \includegraphics[width=1\textwidth]{mlepropest}
+  \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

likelihood/exercises/mlepropest.m

@@ -18,8 +18,9 @@ for i = 1:length(sigmas)
     subplot(2, 2, 2*(i-1)+j);
     bins = [1.9:0.005:2.1];
     hist(slopes, bins);
+    xlabel('estimated slope');
     title(sprintf('sigma=%g, n=%d', sigma, n));
   end
 end
 
-savefigpdf(gcf, 'mlepropest.pdf', 12, 7);
+savefigpdf(gcf, 'mlepropest.pdf', 15, 10);

likelihood/exercises/mlestd.m

@@ -32,4 +32,4 @@ for k = 1:length(ns)
     xlabel('standard deviation')
     ylabel('log likelihood')
 end
-savefigpdf(gcf, 'mlestd.pdf', 15, 5);
+savefigpdf(gcf, 'mlestd.pdf', 15, 10);