\caption{$R$ leniency of the decision maker, $T$ is a binary decision, $Y$ is the outcome that is selectively labled. Background features $X$ for a subject affect the decision and the outcome. Additional background features $Z$ are visible only to the decision maker in use. }\label{fig:model}
\end{figure}
We model the selective labels setting as summarized by Figure~\ref{fig:model}\cite{lakkaraju2017selective}.
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@@ -263,13 +244,17 @@ We use a propensity score framework to model $X$ and $Z$: they are assumed conti
%\acomment{We need to start by noting that with a simple example how we assume this to work. If X indicates a safe subject that is jailed, then we know that (I dont know how this applies to other produces) that Z must have indicated a serious risk. This makes $Y=0$ more likely than what regression on $X$ suggests.} done by Riku!
\acomment{I do not understand what we are doing from this section. It needs to be described ASAP.}
%\acomment{I do not understand what we are doing from this section. It needs to be described ASAP.}
Our approach is based on the fact that in almost all cases, some information regarding the latent variable is recoverable. For illustration, let us consider defendant $i$ who has been given a negative decision $\decisionValue_i =0$. If the defendant's private features $\featuresValue_i$ would indicate that this subject would be safe to release, we could easily deduce that the unobservable variable $\unobservableValue_i$ indicated high risk since
%contained so significant information that
the defendant had to be jailed. In turn, this makes $Y=0$ more likely than what would have been predicted based on $\featuresValue_i$ alone.
In an opposite situation, where the features $\featuresValue_i$ clearly imply that the defendant is dangerous and is subsequently jailed, we do not have that much information available on the latent variable.
\acomment{Could emphasize the above with a plot, x and z in the axis and point styles indicating the decision.}
\acomment{The above assumes that the decision maker in the data is good and not bad.}
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@@ -354,6 +339,27 @@ In practise, once we have used Stan, we have $S$ samples from all of the paramet
\caption{$R$ leniency of the decision maker, $T$ is a binary decision, $Y$ is the outcome that is selectively labled. Background features $X$ for a subject affect the decision and the outcome. Additional background features $Z$ are visible only to the decision maker in use. }\label{fig:model}
\end{figure}
\begin{algorithm}
%\item Potential outcomes / CBI \acomment{Put this in section 3? Algorithm box with these?}
\DontPrintSemicolon
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@@ -376,6 +382,8 @@ Using Stan, draw $S$ samples of the all parameters from the posterior distributi
% If X has multiple dimensions or the relationships between the features and the outcomes are clearly non-linear the presented approach can be extended to accomodate non-lineairty. Jung proposed that... Groups... etc etc.
