From e4f3a4439086dfd6c3ddfad0b51b17f170784640 Mon Sep 17 00:00:00 2001
From: Antti Hyttinen <ajhyttin@gmail.com>
Date: Mon, 10 Jun 2019 13:33:16 +0300
Subject: [PATCH] Few comments.
---
paper/sl.tex | 6 ++++--
1 file changed, 4 insertions(+), 2 deletions(-)
diff --git a/paper/sl.tex b/paper/sl.tex
index 00014b4..42be210 100755
--- a/paper/sl.tex
+++ b/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}
--
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