We consider data recorded from a decision making process with the following characteristics~\cite{lakkaraju2017selective}.
We consider data recorded from a decision making process with the following characteristics~\cite{lakkaraju2017selective}.
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Each case is decided by a decision maker and we use$\judge$as an index to the decision maker the case is assigned to.
Each case is decided by a decision maker and let$\judge$ index the decision maker the case is assigned to.
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For each such assignment, a decision maker $\human_\judgeValue$ (where $\judgeValue$ is a particular value for $\judge$) considers a case described by a set of features \allFeatures and makes a binary decision $\decision\in\{0, 1\}$, nominally referred to as {\it positive} ($\decision=1$) or {\it negative} ($\decision=0$).
For each case, described by a set of features \allFeatures, the assigned decision maker $\human_\judgeValue$ (where $\judgeValue$ is a particular value for $\judge$) makes a binary decision $\decision\in\{0, 1\}$, nominally referred to as {\it positive} ($\decision=1$) or {\it negative} ($\decision=0$).
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Intuitively, in our bail-or-jail example of Section~\ref{sec:introduction}, $\human_\judgeValue$ corresponds to the human judge deciding whether to grant bail or not (positive or negative decision, respectively).
In our bail-or-jail example of Section~\ref{sec:introduction}, $\human_\judgeValue$ corresponds to the judge deciding whether to grant bail or not (positive or negative decision, respectively).
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The decision is followed with a binary outcome $\outcome$, which is nominally referred to as {\it successful} ($\outcome=1$) or {\it unsuccessful} ($\outcome=0$).
The decision is followed with a binary outcome $\outcome$, which is nominally referred to as {\it successful} ($\outcome=1$) or {\it unsuccessful} ($\outcome=0$).
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@@ -23,9 +23,9 @@ Back in our example, the decision of the judge is unsuccessful only if the judge
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@@ -23,9 +23,9 @@ Back in our example, the decision of the judge is unsuccessful only if the judge
Otherwise, if the decision of the judge was to keep the defendant in jail, the outcome is by default successful since there can be no bail violation.
Otherwise, if the decision of the judge was to keep the defendant in jail, the outcome is by default successful since there can be no bail violation.
For each case a record $(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)$ is produced 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.
For each case, a record $(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)$ is produced that contains only 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 harshness of the possible crime) and $\unobservable$ corresponds to features that are observed by the judge but do not appear on record (e.g., exact verbal response of the defendant in court).
In our example, $\obsFeatures$ corresponds to publicly recorded information about the bail-or-jail case (e.g., the charged crime) and $\unobservable$ corresponds to features that are observed by the judge but do not appear on record (e.g., exact verbal response of the defendant in court).
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The set of records $\dataset=\{(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)\}$%produced by decision maker \human
The set of records $\dataset=\{(\judgeValue, \obsFeaturesValue, \decisionValue, \outcomeValue)\}$%produced by decision maker \human
comprises what we refer to as the {\bf dataset}.
comprises what we refer to as the {\bf dataset}.
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Figure~\ref{fig:causalmodel} shows the causal diagram of this decision making process.
Figure~\ref{fig:causalmodel} shows the causal diagram of this decision making process.
Based on the recorded data, we wish to evaluate a decision maker \machine that considers a case from the dataset -- and makes its own binary decision $\decision$ based on the recorded features $\obsFeatures$.
We wish to evaluate a decision maker \machine that considers a case from the dataset -- and makes its own binary decision $\decision$ based on the recorded features $\obsFeatures$.
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%, followed by a binary outcome $\outcome$.
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In our example, \machine corresponds to a machine-based automated decision making system that is considered for bail-or-jail decisions.
In our example, \machine corresponds to a machine-based system that is considered for bail-or-jail decisions.
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% Notice that we assume \machine has access only to some of the features that were available to \human, to model cases where the system would use only the recorded features and not other ones that would be available to a human judge.
% Notice that we assume \machine has access only to some of the features that were available to \human, to model cases where the system would use only the recorded features and not other ones that would be available to a human judge.
