@@ -55,13 +55,14 @@ Otherwise, if the decision of the judge was to keep the defendant in jail ($\dec
%\end{equation}
%Antti I think this formula is mostly misleading
In the likely case that a dataset includes decision of many decision makers we mark $\human=\{\human_\judgeValue\}$ and use
$\judge$ as an index to the decision maker the subject is assigned.
$\judge$ as an index to the decision maker the subject is assigned. This allows to in particular us to model decision makers differing in their leniency, the portion of subjects they make a positive decision for.
The product of this process is a record $(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)$, that contains only observations on a subset $\obsFeatures\subseteq\allFeatures$ of the features of the case, the decision $\decision$ of the judge and the outcome $\outcome$ -- but leaves no trace for a subset $\unobservable=\allFeatures\setminus\obsFeatures$ of the features.
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Intuitively, in our example, $\obsFeatures$ corresponds to publicly recorded information about the bail-or-jail case decided by the judge (e.g., the gender and age of the defendant) and $\unobservable$ corresponds to features that are observed by the judge but do not appear on record (e.g., whether the defendant appeared anxious in court).
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The set of records $\dataset=\{(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)\}$ produced by decision maker \human becomes part of what we refer to as the {\bf dataset}.
The set of records $\dataset=\{(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)\}$%produced by decision maker \human
becomes part of what we refer to as the {\bf dataset}.
% -- and the dataset generally includes records from \emph{more than one} decision makers, indexed by $\judgeValue$.
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Figure~\ref{fig:causalmodel} shows the causal diagram of this decision making process.
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@@ -82,7 +83,7 @@ In our example, \machine corresponds to a machine-based automated-decision syste
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The definitions and semantics of decision $\decision$ and outcome $\outcome$ follow those of the first process.
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Moreover, decision maker \machine is also associated with a leniency level $\leniency$, defined as before for \human.
%Moreover, decision maker \machine is also associated with a leniency level $\leniency$, defined as before for \human.
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%The causal diagram for decision-maker \machine is the same as that for \human (Figure~\ref{fig:causalmodel}), except that \machine does not observe variables $\unobservable$.
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@@ -96,6 +97,9 @@ A good decision maker achieves as low failure rate \failurerate as possible.
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Note, however, that a decision maker that always makes a negative decision $\decision=0$, has failure rate $\failurerate=0$, by definition.
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%To produce sensible evaluation of decision maker at varying leniency levels.
%Moreover, decision maker \machine is also associated with a leniency level $\leniency$, defined as before for \human.
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Thus the evaluation to be meaningful, we evaluate decision makers at the different leniency levels $\leniency$.
%Ultimately, our goal is to obtain an estimate of the failure rate \failurerate for a decision maker \machine.
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@@ -111,11 +115,11 @@ In particular, let us consider the case where we wish to evaluate decision maker
Suppose also that the decision in the data was $\decision=0$, in which case the outcome is always positive, $\outcome=1$.
Suppose also that the decision in the data was negative, $\decision=0$, in which case the outcome is always positive, $\outcome=1$.
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If the decision by \machine is $\decision=1$, then it is not possible to tell directly from the dataset what its outcome $\outcome$ would be.
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The approach we take to deal with this challenge is to use counterfactual reasoning to infer $\outcome$(see Section~\ref{sec:imputation} below).
The approach we take to deal with this challenge is to use counterfactual reasoning to infer $\outcome$ if we had $\decision=1$(see Section~\ref{sec:imputation} below).
%Sometimes, we may have control over the leniency level of the decision maker we evaluate.
