Comments about beta_z=0 and epsilon

analysis_and_scripts/notes.tex

@@ -67,6 +67,8 @@
 
 \maketitle
 
+\section*{Contents}
+
 \tableofcontents
 
 \begin{abstract}
@@ -317,27 +319,29 @@ Causal model, ep 	& 0.001074039 	& 0.0414928\\
 
 \subsection{$\beta_Z=0$ and data generated with unobservables.}
 
-If we assign $\beta_Z=0$, almost all failure rates drop to zero in the interval 0.1, ..., 0.3 but the human evaluation failure rate. Figures are drawn in Figures \ref{fig:betaZ_1_5} and \ref{fig:betaZ_0}.
+If we assign $\beta_Z=0$, almost all failure rates drop to zero in the interval 0.1, ..., 0.3 but the human evaluation failure rate. Results are presented in Figures \ref{fig:betaZ_1_5} and \ref{fig:betaZ_0}.
+
+The differences between figures \ref{fig:results_without_Z} and \ref{fig:betaZ_0} could be explained in the slight difference in the data generating process, namely the effect of $W$ or $\epsilon$. The effect of adding $\epsilon$ (noise to the decisions) is further explored in section \ref{sec:epsilon}.
 
 \begin{figure}[H]
     \centering
     \begin{subfigure}[b]{0.5\textwidth}
         \includegraphics[width=\textwidth]{sl_with_Z_4iter_betaZ_1_5}
-        \caption{$\beta_Z=1.5$}
+        \caption{With unobservables, $\beta_Z$ set to 1.5 in algorithm \ref{alg:data_with_Z}.}
         \label{fig:betaZ_1_5}
     \end{subfigure}
     ~ %add desired spacing between images, e. g. ~, \quad, \qquad, \hfill etc. 
       %(or a blank line to force the subfigure onto a new line)
     \begin{subfigure}[b]{0.5\textwidth}
         \includegraphics[width=\textwidth]{sl_with_Z_4iter_beta0}
-        \caption{$\beta_Z=0$}
+        \caption{With unobservables, $\beta_Z$ set to 0 in algorithm \ref{alg:data_with_Z}.}
         \label{fig:betaZ_0}
     \end{subfigure}
     \caption{Effect of $\beta_z$. Failure rate vs. acceptance rate with unobservables in the data (see algorithm \ref{alg:data_with_Z}). Logistic regression was trained on labeled training data. Results from algorithm \ref{alg:perf_comp} with $N_{iter}=4$.}
     \label{fig:betaZ_comp}
 \end{figure}
 
-\subsection{Noise added to the decision and data generated without unobservables}
+\subsection{Noise added to the decision and data generated without unobservables} \label{sec:epsilon}
 
 In this part, Gaussian noise with zero mean and 0.1 variance was added to the probabilities $P(Y=0|X=x)$ after sampling Y but before ordering the observations in line 5 of algorithm \ref{alg:data_without_Z}. Results are presented in Figure \ref{fig:sigma_figure}.