...

paper/imputation.tex

@@ -71,14 +71,14 @@ Taking into account that we need to learn parameters from the data we integrate
 
 
 
+%
+%
+%\subsection{Implementation}
+%
+%Stan allows us to directly sample from the posterior both of the parameters and the unobservable features.
 
 
-\subsection{Implementation}
-
-Stan allows us to directly sample from the posterior both of the parameters and the unobservable features.
-
-
-\subsection{Our approach}
+\subsection{Overview of Our Approach}
 
 Having provided the intuition for our approach, in what follows we describe it in detail.
 %
@@ -107,10 +107,13 @@ In particular, it provides information about the leniency and other parameters o
 
 Our approach for those cases unfolds over three steps: first, it learns a model over the dataset; then, it computes counterfactuals to predict unobserved outcomes; and finally, it uses predictions to evaluate a set of decisions.
 
-%\subsection{Model} \label{sec:model_definition}
+\subsection{The Causal Model} \label{sec:model_definition}
 
 %To make inference we obviously have to learn the parametric model from the data instead of fixed functions of the previous section. We can define the model as a probabilistic due to the simplification of the counterfactual expression in the previous section.
-\spara{Model} The causal diagram of Figure~\ref{fig:causalmodel} provides the structure of causal relationships for quantities of interest.
+
+%\spara{Model}
+
+ The causal diagram of Figure~\ref{fig:causalmodel} provides the structure of causal relationships for quantities of interest.
 We use the following causal model over the observed data, building on what is used by Lakkaraju et al.~\cite{lakkaraju2017selective}. We assume feature vectors $\obsFeaturesValue$ and $\unobservableValue$ representing risk can be consensed to unidimension risk factors, for example by propensity scores. Furthermore, we assume their distribution as Gaussian. Since $Z$ is unobserved we can assume its variance to be 1 without loss of generality, thus $\unobservable \sim N(0,1)$.
 %\begin{eqnarray*}
 %\unobservable &\sim& N(0,1).
@@ -211,8 +214,9 @@ We use prior distributions given in Appendix~X
 %	\prob{\parameters | \dataset} = \frac{\prob{\dataset | \parameters} \prob{\parameters}}{\prob{\dataset}} .
 %\end{equation}
 
+\subsection{Computing Counterfactuals}
 
-\spara{Computing counterfactuals} 
+%\spara{Computing counterfactuals} 
 For the model defined above, the counterfactual $\hat{Y}$ can be computed by the approach of Pearl.
 For fully defined model (fixed parameters)  $\hat{Y}$ can be determined by the following expression:
 \begin{align}
@@ -239,8 +243,8 @@ where \outcome is the outcome recorded in the dataset \dataset.
 
 
 
-\spara{Implementation} 
-The result of Equation~\ref{eq:theposterior}  is computed numerically:
+%\spara{Implementation} 
+The result of Equation~\ref{eq:theposterior}  can be computed numerically:
 %\begin{equation}
 %	\cfoutcome \approxeq \sum_{\parameters\in\sample}\prob{\outcome = 1 | \obsFeatures = \obsFeaturesValue, \decision = 1, \alpha, \beta_{_\obsFeatures}, \beta_{_\unobservable}, \unobservable} \label{eq:expandcf}
 %\end{equation}
@@ -324,7 +328,9 @@ In practice, we use the MCMC functionality of Stan\footnote{\url{https://mc-stan
 %% 
 %The Gaussians were restricted to the positive real numbers and both had mean $0$ and variance $\tau^2=1$ -- other values were tested but observed to have no effect.
 
-\spara{Evaluation of decisions}
+%\spara{Evaluation of decisions}
+
+\subsection{Evaluating Decision Makers}
 Expression~\ref{eq:expandcf} gives us a direct way to evaluate the outcome of decisions $\decision_\machine$ for any data entry for which $\decision_\human = 0$.
 %
 Note though that, unlike entries for which $\decision_\human = 1$ that takes integer values $\{0, 1\}$, \cfoutcome may take fractional values $\cfoutcome \in [0, 1]$.