From 75c2fe4dd6f8c9c7cd8df8d08c83c6b0070c5986 Mon Sep 17 00:00:00 2001
From: Riku-Laine <28960190+Riku-Laine@users.noreply.github.com>
Date: Fri, 31 May 2019 13:49:02 +0300
Subject: [PATCH] fig 2 caption
---
paper/sl.tex | 8 ++++----
1 file changed, 4 insertions(+), 4 deletions(-)
diff --git a/paper/sl.tex b/paper/sl.tex
index 933ee6f..d785027 100755
--- a/paper/sl.tex
+++ b/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}
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