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do(S=s)

A S=s’ A do(S=s) S s
S s A
B B B s Y Y s S W Y
S Y Y Y s Y=y Y =y’ = { → → → }
s
(a) (b) (c) (d)

Figure 8. (a) The tony causal graph; (b) the counterfactual graph of (a); (c) the W-graph; and (d) the “kite” graph.

For semi-Markovian models, the causal effect ( | ( )) is not always identifiable, hence the total effect is
not always identifiable. The causal effect ( | ( )) is identifiable if and only if it can be reduced to a -free
expression (i.e., turning the intervention operator ( ) to observational probabilities) by -calculus [10] . -
calculus is composed of three inference rules: (i) insertion/deletion of observations, i.e., ( |z, ) = ( |z)
providedthat and aredependence-separatedatfixed andZafterallarrowsleadingto havebeendeleted
in causal graph; (ii) action/observation exchange, i.e., ( | ( ),z) = ( | ,z) if and are probabilistically
conditionally independent at fixed Z after deleting all arrows starting from Z in causal graph; and (iii) deletion
of actions, i.e., ( | ( )) = ( ) if there are no causal paths between and .

-calculus has been proven to be complete, that is, it is sufficient to derive all identifiable causal effects by -
calculus [100] . However, it is difficult to determine the correct order of application of these rules, and the wrong
order may misjudge the identifiability of causal effects or produce a very complex expression. To address this
issue, several studies attempt to give the explicit graphical criteria and map them to simple and concise -free
expressions [101,104] . A simple case of the identifiability of the causal effect ( | ( )) is when the sensitive
attribute is not influenced by any confounder [105] . In other words, the causal effect of is identifiable, if all
parents of are observable. Graphically, there is no bi-directional edge connected to . Formally, the causal
effect ( | ( )) can be computed as follows:
∑
( | ( )) = ( | , ( )) (pa( )) (18)
pa( )
where ( ) represents the values of parent variables of .

A complex case where the causal effect of on V = V \ { } is identifiable is that there may exist a bi-directed
′
edge connected to the sensitive attribute , but there are no hidden confounders connected to any direct child
of [105] . Graphically, there is no bi-directional edge connected to any child of . If such criterion is satisfied,
the causal effect (v | ( )) is identifiable and is given by:
′
∏ ∑ (v )
′
′ (19)
(v | ( )) = ( ( |pa( ))) ∏
( |pa( ))
∈ ℎ( ) ∈ ℎ( )
where ℎ( ) denotes the set of ’s children and pa( ) denotes the set of values of ’s parents. Equation (19)
can be easily adapted to assess the effect of the sensitive attribute on outcome .
Tianetal. [105] alsofoundthat, although (v | ( )) isnotidentifiable, (w| ( )) isstillidentifiableforsome
′
subsets W of V. Specifically, causal effect (w| ( )) is identifiable if there is no bi-directed path connecting
to any of its direct children in G (W), where (W) denotes the union of a set W and the set of ancestors
of the variables in W. G (W) denotes the sub-graph of G composed only of variables in (W).

Moreover, the causal effect of sensitive attribute on outcome ( | ( )) can also be computed by using the
front-door criterion, if ( | ( )) is identifiable. Before introducing the front-door criterion, the concepts of