Counterfactual quantities

Using structural model semantics, Pearl gives definitions of six counterfactual quantities.

Let $Y_a$ be a random variable representing a binary outcome $Y$ after a binary treatment $X$ is forced to value $a$ ; and let $x$ and $y$ denote events $X=1$ and $Y=1$ , and $x', y'$ their complements.

Probability that the absence of event $x$ would prevent event $y$ , given that in fact $x$ and $y$ did occur. — attribution

$P(Y_0=0\vert x, y)$

Probability that the presence of event $x$ would produce event $y$ , given that in fact $x$ and $y$ did not occur. — susceptibility

$P(Y_1=1\vert x^{\prime}, y^{\prime})$

Probability that the absence of event $x$ would prevent event $y$ and the presence of event $x$ would produce event $y$ . — necessary and sufficient cause

$P(Y_0=0, Y_1=1)$

Probability that the absence of event $x$ would prevent event $y$ , given that in fact $y$ did occur. — proportion of observed cases in the total population that would not have occurred had the population been entirely unexposed (disablement)

$P(Y_0=0\vert y)$

Probability that the presence of event $x$ would produce event $y$ , given that in fact $y$ did not occur. — enablement

$P(Y_1=1\vert y^{\prime})$

Probability that the presence of event $x$ would produce event $y$ , given that in fact $x$ did not occur. — Effect of treatment on the treated

$P(Y_1=1\vert x^{\prime})$

by Boris Sobolev/in Counterfactual theory/on June 06, 2012