Merge branch 'master' of https://version.helsinki.fi/rikulain/bachelors-thesis

paper/sl.tex

@@ -33,7 +33,7 @@
 \usepackage{footnote} % show footnotes in tables
 \makesavenoteenv{table}
 
-
+\newcommand{\antti}[1]{{{\color{orange} [AH: #1]}}}
 
 \newcommand{\ourtitle}{A Causal Approach for Selective Labels}
 
@@ -168,6 +168,8 @@ We wish to calculate the probability of undesired outcome (\outcome = 0) at a fi
 & = \sum_\featuresValue \prob{\outcome = 0 | \decision = 1, \features = \featuresValue} \prob{\decision = 1 | \leniency = \leniencyValue, \features = \featuresValue} \prob{\features = \featuresValue}
 \end{align*}
 
+\antti{Here one can drop do even at the first line according to do-calculus rule 2, i.e. $P(Y=0|do(R=r))=P(Y=0|R=r)$. However, do-calculus formulas should be computed by first learning a graphical model and then computing the marginals using the graphical model. This gives more accurate result. Michael's complicated formula essentially does this, including forcing $P(Y=0|T=0,X)=0$ (the model supports context-specific independence $Y \perp X | T=0$.)}
+
 Expanding the above derivation for model \score{\featuresValue} learned from the data
 \[
 \score{\featuresValue} = \prob{\outcome = 0 | \features = \featuresValue, \decision = 1},
@@ -221,7 +223,7 @@ random variables so that
 \prob{\outcome = 0| \features = \featuresValue} = \dfrac{1}{1+\exp\{-\featuresValue\}}.
 \]
 
-The decision variable $\decision$ was set to 0 if the probability $\prob{\outcome = 0| \features = \featuresValue}$ resided in the top $(1-\leniencyValue)\cdot 100 \%$ of the subjects appointed for that judge.
+The decision variable $\decision$ was set to 0 if the probability $\prob{\outcome = 0| \features = \featuresValue}$ resided in the top $(1-\leniencyValue)\cdot 100 \%$ of the subjects appointed for that judge. \antti{How was the final Y determined? I assume $Y=1$ if $T=0$, if $T=1$ $Y$ was randomly sampled from $\prob{\outcome| \features = \featuresValue}$ above? Delete this comment when handled.}
 Results for estimating the causal quantity $\prob{\outcome = 0 | \doop{\leniency = \leniencyValue}}$ with various levels of leniency $\leniencyValue$ under this model are presented in Figure \ref{fig:without_unobservables}.
 
 \begin{figure}