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(a) (b) (c)

Figure 3. (a) An example causal graph based on Markovian assumption; (b) a causal graph based on semi-Markovian assumption; and (c)
a causal graph after performing an intervention on .

pointing to variable . Figure 3(c) shows the causal diagram after the intervention ( ). The mathematical
meaning of ( ) is defined as the substitution of equation = ( ( ), ) with = . For another
endogenous variable which is affected by the intervention, its post-intervention distribution under ( )
is denoted by ( | ( )) or ( ) for short. Intuitively, ( | = ) represents the population distribution
of condition on observing attribute value of individuals is , while ( | ( = )) (i.e., ( | ( )))
represents the population distribution of if everyone in the population had their value fixed at . This
post-intervention distribution ( | ( )) is considered a counterfactual distribution since the intervention
( ) forces to take a certain value different from the one it would take in the actual world. For exam-
ple, if represents sex ( , male; , female) and represents the hiring decision ( , hired; , not hired),
+
+
−
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( | = , = ) involves two worlds: a real world that a male applicant has been hired and a counterfac-
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tual world where the same applicant is female. Such expression means that, when a job-hunter whose gender
is male has been observed to be hired, what is the probability that the same job-hunter would still be hired if
this job-hunter were female.
Causality-based fairness notions aim to tell whether the outcome of a decision made by the AI decision model
is discriminative, which are expressed by interventional and counterfactual probability distributions. The ap-
plication of the causality-based fairness notions not only requires a dataset D as input but also relies on a
causal graph G. Causal approaches aim to limit the causal effects of sensitive attributes on decisions, which
are computed by interventional and counterfactual probabilities. However, since these probabilities cannot be
observed, they fail to be uniquely assessed from D and G in some cases, which is called the unidentifiable issue.
In other words, if two variables have different causal effect measures resulting from different causal models
that all agree with the observational distribution, the causal effects are unidentifiable.

4. CAUSALITY-BASED FAIRNESS NOTIONS
Pearl defined the causality as three rungs: correlation, intervention, and counterfactual [25] . The first rung
(correlation) reflects the ability of observation, which aims to discover the patterns in the environment. The
secondrung(intervention)reflectstheabilityofaction,whichreferstothepredictionoftheresultsofdeliberate
changes to the environment. The third rung (counterfactual) refers to the ability to imagine the counterfactual
world and speculate on the causes of the observed phenomena. The second and third rungs aim to expose
the root causes of the patterns that we observe. Thus far, many fairness metrics have been proposed and all of
them can be placed in such causal rungs. Figure 4 presents categorization of fairness notions. Early definitions
of fairness are based on statistical correlations, all of which can be found at the first rung. All causality-based
fairness notions can be found at the second and third rungs, each of which considers the mechanism of data
generation. Thus, causality-based fairness notions can better reveal the causes of discrimination than statistics-
based ones and have been attracting more and more attention.