notes.tex

  \caption{Summary of modules (under construction)}
  \begin{tabular}{lll}
    \toprule
    \multicolumn{3}{c}{Module type} \\[.5\normalbaselineskip]
    \textbf{Data generator} & \textbf{Decider} & \textbf{Evaluator}  \\
    \midrule
    {\ul Without unobservables}	& Independent decisions		& {\ul Labeled outcomes} \\
     					 	& 1. draw T from a Bernoulli	& \tabitem Data $\D$ with properties $\{x_i, t_i, y_i\}$ \\
    {\ul With unobservables}       	& with $P(T=0|X, Z)$			& \tabitem acceptance rate r \\
    \tabitem $P(Y=0|X, Z, W)$ 	& 2. determine with $F^{-1}(r)$	& \tabitem knowledge that X affects Y \\[.5\normalbaselineskip]
    
     {\ul With unobservables}	& Non-independent decisions  	& {\ul True evaluation} \\
     \tabitem assign Y by		& 3. sort by $P(T=0|X, Z)$		& \tabitem Data $\D$ with properties $\{x_i, t_i, y_i\}$ \\
     "threshold rule"			& and assign $t$ by $r$  		& and \emph{all outcome labels} \\
     						&   						& \tabitem acceptance rate r \\
     						&   						& \tabitem knowledge that X affects Y \\[.5\normalbaselineskip]
     
     &  & {\ul Human evaluation} \\
     &  & \tabitem Data $\D$ with properties $\{x_i, j_i, t_i, y_i\}$ \\
     &  & \tabitem acceptance rate r \\[.5\normalbaselineskip]
     
     &  & {\ul Contraction algorithm} \\
     &  & \tabitem Data $\D$ with properties $\{x_i, j_i, t_i, y_i\}$ \\
     &  & \tabitem acceptance rate r \\
     &  & \tabitem knowledge that X affects Y \\[.5\normalbaselineskip]
     
     &  & {\ul Causal model} \\
     &  & \tabitem Data $\D$ with properties $\{x_i, t_i, y_i\}$ \\
     &  & \tabitem acceptance rate r \\
     &  & \tabitem knowledge that X affects Y \\[.5\normalbaselineskip]
     
     &  & {\ul Monte Carlo evaluator} \\
     &  & \tabitem Data $\D$ with properties $\{x_i, j_i, t_i, y_i\}$ \\
     &  & \tabitem acceptance rate r \\
     &  & \tabitem knowledge that X affects Y \\
     &  & \tabitem more intricate knowledge about $\M$ ? \\[.5\normalbaselineskip]
    \bottomrule
  \end{tabular}
  \label{tab:modules}
\end{table}

\begin{thebibliography}{9} % Might have been apa

\bibitem{dearteaga18}
   De-Arteaga, Maria. Learning Under Selective Labels in the Presence of Expert Consistency. 2018. 
\bibitem{lakkaraju17} 
   Lakkaraju, Himabindu. The Selective Labels Problem: Evaluating Algorithmic Predictions in the Presence of Unobservables. 2017. 

\end{thebibliography}

\end{document}