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evaluate the causal effect of on (more details of intervention can be seen in Section 3). Figure 2(b) shows
the causal structure of such an example after intervening on . The hiring decisions made by the company are
fair, if the hiring proportions when all applicants in the population have religious beliefs are the same as the
hiring proportions when all applicants in the population have no religious beliefs, i.e., ( = 1| ( = 1)) =
( = 1| ( = 0)). Formally, these probabilities in this example are obtained as below:
∑
( = 1| ( = 1)) = ( = 1| = 1, = ) · ( = )
∈{0,1}
= 0.02 × 0.5 + 0.25 × 0.5 = 0.135
∑
( = 1| ( = 0)) = ( = 1| = 0, = ) · ( = )
∈{0,1}
= 0.03 × 0.5 + 0.24 × 0.5 = 0.135

These values confirm that the hiring decisions made by the company do not discriminate against applicants
with religious beliefs. Therefore, it is critical to conduct a causal analysis of the problem, since understanding
thecausal mechanismsbehindtheproblemcannotonlyhelptodetect discriminationbutalso helptointerpret
the sources of discrimination.

3. PRELIMINARIES AND NOTATION
In this review, an attribute is denoted by an uppercase letter, e.g., ; a subset of attributes is denoted by a bold
uppercase letter, e.g., X; a domain value of attribute is denoted by a lowercase letter, e.g., ; and the value
assignment of subset attributes X is denoted by a bold lowercase letter, e.g., x. In particular, represents the
sensitive attribute (e.g., race) and represents the predicted result of the AI model system (e.g., loans).

One of the most popular causal model frameworks is Pearl’s Structural Causal Model (SCM) [10] . A structural
causal model M is represented by a quadruple ⟨U, (U),V,F⟩:
1. U denotes exogenous variables that cannot be observed but constitute the background knowledge behind
the model.
2. (U) represents the joint probability distribution of U.
3. V denotes endogenous variables that can be observed.
4. F denotes a set of functions mapping from U ∪ V to V, which reflects the causal relationship between
variables. For each ∈ V, there is a mapping function ∈ F from U ∪ (V \ ) to , i.e., =
( ( ), ), where parent variables ( ) ⊂ V \ are the endogenous variables that directly control
the value of , and is a set of exogenous variables that directly determine .

A causal model M is associated with a causal graph G = ⟨V, E⟩, a directed acyclic graph, where V is a set
of nodes, each of which represents an endogenous variable of V, and each element in E indicates a directed
edge →, pointing from a node ∈ U ∪ V to another node ∈ V if uses values of as input, which
represents a causal relationship between the corresponding variables. The exogenous variables U can be either
independent or dependent. If the exogenous variables U are mutually independent, the causal model is called
Markovian model. In this case, exogenous variables typically are not represented in the causal diagram. In
the case, the exogenous variables are mutually dependent (hidden confounders), the causal model is called
the semi-Markovian model. In the semi-Markovian model, dashed bi-directed edges are used to represent the
hidden confounders between two variables. Figure 7 shows examples of causal graphs of Markovian model
[Figure 3(a)] and semi-Markovian model [Figure 3(b)].

An intervention simulates the physical interventions that force some variable to take certain values regard-
less of the corresponding function , denoted by ( ). In the causal graph, it is shown as discarding all edges