From 617637c3c44b1457593101d597c717cc0f2ff02d Mon Sep 17 00:00:00 2001
From: Antti Hyttinen <ajhyttin@gmail.com>
Date: Thu, 16 Jan 2020 16:59:32 +0200
Subject: [PATCH] leniency is
---
paper/imputation.tex | 6 +++---
1 file changed, 3 insertions(+), 3 deletions(-)
diff --git a/paper/imputation.tex b/paper/imputation.tex
index c236852..5ea43b9 100644
--- a/paper/imputation.tex
+++ b/paper/imputation.tex
@@ -137,8 +137,8 @@ Since the decision are ultimately based on the risk factors for behaviour, we mo
\begin{equation}
\prob{\decision_{\human} = 0~|~\leniencyValue_\human,\obsFeaturesValue, \unobservableValue} = \invlogit(\alpha_\human + \gamma_\obsFeaturesValue\obsFeaturesValue + \gamma_\unobservableValue \unobservableValue + \epsilon_\decisionValue) \label{eq:judgemodel}
\end{equation}%
-Note that index $j$ refers to decision maker $\human$. Parameter $\alpha_\human$ provides for the leniency of a decision maker by $\logit(\leniencyValue_\human)$.
-Note that we are making the simplifying assumption that coefficients $\gamma$ are the same for all defendants, but decision makers are allowed to differ in intercept $\alpha_j \approx \logit(\leniencyValue_j)$ so as to model varying leniency levels among them. % (Eq. \ref{eq:leniencymodel}).
+Intercept $\alpha_\human$ provides for the leniency of a decision maker $\human$ by $\logit(\leniencyValue_\human)$.
+Note that we are making the simplifying assumption that coefficients $\gamma$ are the same for all defendants, but decision makers are allowed to differ in intercept $\alpha_\human \approx \logit(\leniencyValue_\human)$ so as to model varying leniency levels among them. % (Eq. \ref{eq:leniencymodel}).
%The decision makers in the data differ from each other only by leniency.
%\noindent
@@ -219,7 +219,7 @@ We use prior distributions given in Appendix~X to ensure the identifiability of
%\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:
+For fully defined model (fixed parameters) the counterfactual expectation can be determined by the following expression:
%\begin{align}
%\cfoutcome & = \int \prob{\outcome=1|\decision=1,\obsFeaturesValue,\unobservableValue} \prob{z|\leniency=\leniencyValue, \decision_{\human} =0, \obsFeaturesValue}\diff{\unobservableValue}
%\end{align}
--
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