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\section{Related work}
\label{sec:related}

In this paper we considered the overall setting as formulated by~\citet{lakkaraju2017selective}, and building on~\cite{Jung2,mccandless2007bayesian,dearteaga2018learning}, showed that causally informed counterfactual imputation can achieve accurate results.
%In addition to Lakkaraju et al.~\citet{lakkaraju2017selective} which we build upon, several papers consider related problems to ours. 
Note that our setting allowing for unobserved confounding does not fulfill the ignorability or missing at random (MAR), preventing the use of methods assuming them~\cite{lakkaraju2017selective,DBLP:conf/icml/DudikLL11,bang2005doubly,little2019statistical}.

De-Arteaga et al. also note the possibility of using decision in the data to correct for selective labels, assuming expert consistency~\cite{dearteaga2018learning}. They directly impute decisions as outcomes and consider learning automatic decision makers from this augmented data. In contrast, our approach on decision maker evaluation is based on a rigorous probabilistic model accounting for different leniencies and unobservables. Furthermore, our approach gives accurate results even with random decision makers that clearly violate the expert consistency assumption.

In reinforcement learning a similar scenario is consider as offline policy evaluation, where the objective is to learn from 
data recorded under some policy, the goodness of some other policies \cite{Jung2,DBLP:conf/icml/ThomasB16}. In particular, Jung et al. \cite{Jung2,jung2018algorithmic} consider sensitivity analysis in a similar scenario as ours, but without directly modelling judges with multiple leniencies.
Mc-Candless et al. perform Bayesian sensitivity analysis on the priors while taking into account latent confounding~\cite{mccandless2007bayesian}.
\cite{kallus2018confounding} obtain improved policies from data possibly biased by a baseline policy.


\citet{mccandless2017comparison} do what?

\citet{kleinberg2018human} present in detail discussion of employing contraction and a particular type of imputation

To properly assess decision procedures for their performance and fairness we need to understand the causal relations \cite{DBLP:conf/icml/NabiMS19,DBLP:conf/icml/Kusner0LS19}

More applied work includes works such as~\cite{murder,tolan2019why}.

\cite{madras2019fairness} learn fair and acccurate treatment policies from biased data   \acomment{They cite Lakkaraju, De arteaga as conceptually similar work.}

\cite{coston2020counterfactual} propose counterfactual measures for performance  metrics with doubly robust estimation of these metrics. The first assumes absense of unobserved variables.

An issue we do not discuss in this paper is fairness  \citep{DBLP:conf/icml/Kusner0LS1,madras2019fairness,corbett2017algorithmic}.

Literature on risks and redicivism? \cite{chouldechova2017fair}


%difference on jung and mcCandless
%they have u as binary, no it is normal there are many
%different groups maybe have diffe

%How do they actually differ.
%\acomment{We should refer to Deartega somewhere early on, they have made the same discovery as we put presented it poorly.}


Recent research has shown the value of counterfactual reasoning in similar setting as this paper, for  fairness of decision making, and applications in online advertising~\cite{DBLP:journals/jmlr/BottouPCCCPRSS13,DBLP:conf/icml/Kusner0LS19,DBLP:conf/icml/NabiMS19,DBLP:conf/icml/JohanssonSS16,pearl2000}. 

missing data refs? This \cite{little2019statistical}???

Some imputation book?


%\subsection{Imputation}







