fig 2 caption

paper/sl.tex

@@ -210,18 +210,18 @@ Every judge's leniency rate $\leniency$ was sampled uniformly from a half-open i
 Private features $\features$ were defined as i.i.d standard Gaussian random variables.
 Next, probabilities for negative results $\outcome = 0$ were calculated as 
 \[
-\prob{\outcome = 0| \features = \featuresValue} = \dfrac{1}{1+\exp\{-\featuresValue\}}
+\prob{\outcome = 0| \features = \featuresValue} = \dfrac{1}{1+\exp\{-\featuresValue\}} = p_{y_0}
 \]
 
-and consequently $\outcome \sim \text{Bernoulli}(1 - \prob{\outcome = 0| \features = \featuresValue})$.
-The decision variable $\decision$ was set to 0 if the value $\prob{\outcome = 0| \features = \featuresValue}$ resided in the top $(1-\leniencyValue)\cdot 100 \%$ of the subjects appointed for that judge.
+and consequently $\outcome \sim \text{Bernoulli}(1 - p_{y_0})$.
+The decision variable $\decision$ was set to 0 if the value $p_{y_0}$ resided in the top $(1-\leniencyValue)\cdot 100 \%$ of the subjects appointed for that judge.
 Results for estimating the causal quantity $\prob{\outcome = 0 | \doop{\leniency = \leniencyValue}}$ with various levels of leniency $\leniencyValue$ are presented in Figure \ref{fig:without_unobservables}.
 
 \begin{figure}
 \begin{center}
 \includegraphics[width=\columnwidth]{img/without_unobservables.png}
 \end{center}
-\caption{$\prob{\outcome = 0 | \doop{\leniency = \leniencyValue}}$ with varying levels of acceptance rate. Error bars denote standard error of the mean across simulations.}
+\caption{$\prob{\outcome = 0 | \doop{\leniency = \leniencyValue}}$ with varying levels of acceptance rate. Error bars denote standard error of the mean.}
 \label{fig:without_unobservables}
 \end{figure}