diff --git a/paper/sl.tex b/paper/sl.tex index 5724bad42d8a6d3d18f6f7b0ca4f22708bb3428d..4d7cca1d798e529f960cfaf8b3ba0f88f8588f7a 100755 --- a/paper/sl.tex +++ b/paper/sl.tex @@ -227,26 +227,7 @@ This estimate is vital in the employment machine learning and AI systems to ever \subsection{Causal Modeling} -\begin{figure} - \begin{tikzpicture}[->,>=stealth',node distance=1.5cm, semithick] - - \tikzstyle{every state}=[fill=none,draw=black,text=black] - \node[state] (R) {$R$}; - \node[state] (X) [right of=R] {$X$}; - \node[state] (T) [below of=X] {$T$}; - \node[state] (Z) [rectangle, right of=X] {$Z$}; - \node[state] (Y) [below of=Z] {$Y$}; - - \path (R) edge (T) - (X) edge (T) - edge (Y) - (Z) edge (T) - edge (Y) - (T) edge (Y); -\end{tikzpicture} -\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}. @@ -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.} @@ -354,6 +339,27 @@ In practise, once we have used Stan, we have $S$ samples from all of the paramet % \end{itemize} %\end{itemize} +\begin{figure} + \begin{tikzpicture}[->,>=stealth',node distance=1.5cm, semithick] + + \tikzstyle{every state}=[fill=none,draw=black,text=black] + + \node[state] (R) {$R$}; + \node[state] (X) [right of=R] {$X$}; + \node[state] (T) [below of=X] {$T$}; + \node[state] (Z) [rectangle, right of=X] {$Z$}; + \node[state] (Y) [below of=Z] {$Y$}; + + \path (R) edge (T) + (X) edge (T) + edge (Y) + (Z) edge (T) + edge (Y) + (T) edge (Y); +\end{tikzpicture} +\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 @@ -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. + + \section{Related work} Discuss this: \cite{DBLP:conf/icml/Kusner0LS19}