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+++ b/Kandi.tex
@@ -75,7 +75,7 @@
 \addtolength{\voffset}{0.45cm}
 \addtolength{\textheight}{-0.9cm}
 
-\title{Kandidaatintutkielma\\ {\Large Kausaalipäättely valikoitumisharha korjaamisessa}} % Parempi otsikko
+\title{Kandidaatintutkielma\\ {\Large Kausaalipäättely valikoitumisharhan korjaamisessa}} % Parempi otsikko
 \author{Riku Laine\\ Valtiotieteellinen tiedekunta \\ Helsingin yliopisto}
 \date{\today}
 
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+++ b/paper/sl.tex
@@ -98,26 +98,31 @@ The decision is based on the following variables. First, the features \features
 Secondly, the leniency of the judge, expressed as a variable \leniency.
 Specifically, we assume that every judge evaluates a given candidate according to the probability 
 \[
-\prob{\outcome = 1 | \features = \featuresValue, \doop{\decision = 1}} 
+\prob{\outcome = 0 | \features = \featuresValue, \doop{\decision = 1}} 
 \]
-that the candidate will violate bail conditions (\outcome = 1) if they were granted bail.
+that the candidate will violate bail conditions (\outcome = 0) if they were granted bail.
 We write \outcome = 0 to refer to the case when the defendant does not violate bail, whether bail is granted or not.
 The \doop{condition} expression signifies that, in evaluating the probability, we consider the event where the condition  (here, it is the condition $\decision = 1$) is imposed to the data-generation process (and therefore alters the generative model).
 In addition, we assume that every judge would assign the same value to the above probability, given by a function \score{\featuresValue}.
 \[
-\score{\featuresValue} = \prob{\outcome = 1 | \features = \featuresValue, \doop{\decision = 1}}
+\score{\featuresValue} = \prob{\outcome = 0 | \features = \featuresValue, \doop{\decision = 1}}
 \]
 The assumption that, essentially, all judges have the same model for the probability that a defendant would violate bail is not far-fetched for the purposes of our analysis, particularly taking into account that \score{\featuresValue} can be learned from the observed data
 \[
-\prob{\outcome = 1 | \features = \featuresValue, \doop{\decision = 1}} = \prob{\outcome = 1 | \features = \featuresValue, \decision = 1}
+\prob{\outcome = 0 | \features = \featuresValue, \doop{\decision = 1}} = \prob{\outcome = 0 | \features = \featuresValue, \decision = 1}
 \]
 and that data are publicly accessible, allowing us to assume that all judges have access to the same information.
 Where judges {\it do differ} is at the level of their leniency \leniency.
 Following the above assumptions, a judge with leniency \leniency = \leniencyValue grants bail to the defendants for which $F(\featuresValue) < r$, where $F$ is the cumulative distribution.
 \begin{equation}
-	F(\featuresValue_0) = \int { \indicator{\prob{\outcome = 1| \decision = 1, \features = \featuresValue} > \prob{\outcome = 1| \decision = 1, \features = \featuresValue_0}}	d\prob{\featuresValue}	}
+	F(\featuresValue_0) = \int { \indicator{\prob{\outcome = 0| \decision = 1, \features = \featuresValue} > \prob{\outcome = 0| \decision = 1, \features = \featuresValue_0}}	d\prob{\featuresValue}	} 
 \end{equation}
 
+which should be equal to
+
+\begin{equation}
+	F(\featuresValue_0) = \int {\prob{\featuresValue}  \indicator{\prob{\outcome = 0| \decision = 1, \features = \featuresValue} > \prob{\outcome = 0| \decision = 1, \features = \featuresValue_0}} d\featuresValue}
+\end{equation}
 The bail-or-jail scenario is just one example of settings that involve a decision $\decision \in\{0,1\}$ that is based on individual features \features and leniency (acceptance rate) \leniency -- and where a behavior of interest \outcome is observed only for the cases where \decision = 1.
 The diagram of the causal model is shown in Figure~\ref{fig:causalmodel}.
 Our results are applicable to other scenarios with same causal model.
@@ -138,23 +143,23 @@ Performance is measured {\it for a given leniency level} as the rate at which ba
 In other words, performance is measured as the probability that a decision lead to undesired outcome.
 \section{Analysis}
 
-We wish to calculate the probability of undesired outcome (\outcome = 1) at a fixed leniency level.
+We wish to calculate the probability of undesired outcome (\outcome = 0) at a fixed leniency level.
 \begin{align*}
-& \prob{\outcome = 1 | \doop{\leniency = \leniencyValue}} = \nonumber \\
-& = \sum_\decisionValue \prob{\outcome = 1, \decision = \decisionValue | \doop{\leniency = \leniencyValue}} \nonumber \\
-& = \prob{\outcome = 1, \decision = 0 | \doop{\leniency = \leniencyValue}} + \prob{\outcome = 1, \decision = 1 | \doop{\leniency = \leniencyValue}} \nonumber \\
-& = 0 + \prob{\outcome = 1, \decision = 1 | \doop{\leniency = \leniencyValue}} \nonumber \\
-& = \prob{\outcome = 1, \decision = 1 | \doop{\leniency = \leniencyValue}} \nonumber \\
-& = \sum_\featuresValue \prob{\outcome = 1, \decision = 1, \features = \featuresValue | \doop{\leniency = \leniencyValue}} \nonumber \\
-& = \sum_\featuresValue \prob{\outcome = 1, \decision = 1 | \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\features = \featuresValue | \doop{\leniency = \leniencyValue}} \nonumber \\
-& = \sum_\featuresValue \prob{\outcome = 1, \decision = 1 | \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\features = \featuresValue} \nonumber \\
-& = \sum_\featuresValue \prob{\outcome = 1 | \decision = 1, \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\decision = 1 | \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\features = \featuresValue} \nonumber \\
-& = \sum_\featuresValue \prob{\outcome = 1 | \decision = 1, \features = \featuresValue} \prob{\decision = 1 | \leniency = \leniencyValue, \features = \featuresValue} \prob{\features = \featuresValue}
+& \prob{\outcome = 0 | \doop{\leniency = \leniencyValue}} = \nonumber \\
+& = \sum_\decisionValue \prob{\outcome = 0, \decision = \decisionValue | \doop{\leniency = \leniencyValue}} \nonumber \\
+& = \prob{\outcome = 0, \decision = 0 | \doop{\leniency = \leniencyValue}} + \prob{\outcome = 0, \decision = 1 | \doop{\leniency = \leniencyValue}} \nonumber \\
+& = 0 + \prob{\outcome = 0, \decision = 1 | \doop{\leniency = \leniencyValue}} \nonumber \\
+& = \prob{\outcome = 0, \decision = 1 | \doop{\leniency = \leniencyValue}} \nonumber \\
+& = \sum_\featuresValue \prob{\outcome = 0, \decision = 1, \features = \featuresValue | \doop{\leniency = \leniencyValue}} \nonumber \\
+& = \sum_\featuresValue \prob{\outcome = 0, \decision = 1 | \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\features = \featuresValue | \doop{\leniency = \leniencyValue}} \nonumber \\
+& = \sum_\featuresValue \prob{\outcome = 0, \decision = 1 | \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\features = \featuresValue} \nonumber \\
+& = \sum_\featuresValue \prob{\outcome = 0 | \decision = 1, \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\decision = 1 | \doop{\leniency = \leniencyValue}, \features = \featuresValue} \prob{\features = \featuresValue} \nonumber \\
+& = \sum_\featuresValue \prob{\outcome = 0 | \decision = 1, \features = \featuresValue} \prob{\decision = 1 | \leniency = \leniencyValue, \features = \featuresValue} \prob{\features = \featuresValue}
 \end{align*}
 
 Expanding the above derivation for model \score{\featuresValue} learned from the data
 \[
-\score{\featuresValue} = \prob{\outcome = 1 | \features = \featuresValue, \decision = 1},
+\score{\featuresValue} = \prob{\outcome = 0 | \features = \featuresValue, \decision = 1},
 \]
 the {\it generalized performance} \generalPerformance of that model is given by the following formula.
 \begin{equation}