More to imputation. Reference in setting.

paper/imputation.tex

@@ -52,7 +52,8 @@ We use the following causal model over this structure, building on what is used
 %
 First, we assume that the
 % the observed feature vectors \obsFeatures and 
-unobserved features \unobservable can be modeled as a one-dimensional risk factor~\cite{mccandless2007bayesian}, for example by using propensity scores~\cite{rosenbaum1983central,austin2011introduction}. 
+unobserved features \unobservable can be modeled as a one-dimensional risk factor~\cite{mccandless2007bayesian,rosenbaum1983central,austin2011introduction}.
+%, for example by using propensity scores~\cite{}. 
 %
 Moreover, we are also going to present our modeling approach for the case of a single observed feature \obsFeatures -- this is done only for simplicity of presentation, as it is straightforward to extend the model to the case of multiple features \obsFeatures, as we do in the experiments (Section~\ref{sec:experiments}).
 %
@@ -77,7 +78,7 @@ Here \invlogit is the standard logistic function.
 %
 % Since the decisions are ultimately based on expected behaviour, 
 
-We model the decisions in the data similarly according to a logistic regression over the features:
+We model the decisions in the data similarly, since the decisions are ultimately based on expected behaviour, according to a logistic regression over the features:
 \begin{equation}
 \prob{\decision = 1~|~\judgeValue,\obsFeaturesValue, \unobservableValue}  =  \invlogit(\alpha_\judgeValue + \gamma_\obsFeatures \obsFeaturesValue + \gamma_\unobservable \unobservableValue 
 )  \label{eq:judgemodel} 
@@ -123,7 +124,7 @@ For a fully defined model (with fixed parameters)  the counterfactual expectatio
 E_{\decision \leftarrow 1}(\outcome|\judgeValue,\decision=0,\obsFeaturesValue)
  &=    \int   \prob{\outcome=1|\decision=1,\obsFeaturesValue,\unobservableValue}  \prob{\unobservableValue|\judgeValue, \decision=0,\obsFeaturesValue} \diff{\unobservableValue} \label{eq:counterfactual_eq}
 \end{align}
-In essence, we determine the distribution of the unobserved features $\unobservable$ using the decision, observed features $\obsFeaturesValue$, and the leniency of the employed decision maker, and then determine the distribution of $\outcome$ conditional on all features, integrating over the unobserved features (see Appendix~\ref{sec:counterfactuals} for more details).
+In essence, we determine the distribution of the unobserved features $\unobservable$ using the decision, observed features $\obsFeaturesValue$, and the leniency of the employed decision maker, and then determine the distribution of $\outcome$ conditional on all features, integrating over the unobserved features (see Appendix~\ref{sec:counterfactuals} for more details). Note that the decision maker model in Equation~\ref{eq:judgemodel} affects the distribution of the unobserved features $\prob{\unobservableValue|\judgeValue, \decision=0,\obsFeaturesValue}$.
 
 Having obtained a posterior probability distribution for parameters \parameters we can estimate the counterfactual outcome value based on the data: 
 \begin{equation}

paper/setting.tex

@@ -10,7 +10,7 @@
 %The setting we consider is described in terms of {\it two decision processes}.
 %We consider a setting where a set of decision makers $\humanset=\{\human_\judgeValue\}$ make decisions for a set of cases.
 %NO WE DO NOT CONSIDER THIS SETTING, THE SETTING IS THAT WE HAVE TO EVALUATE M BASED ON DATA
-We consider data recorded from a decision making process with the following characteristics.
+We consider data recorded from a decision making process with the following characteristics~\cite{lakkaraju2017selective}.
 %
 Each case is decided by one decision maker and we use $\judge$ as an index to the decision maker the case is assigned. 
 %