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necessary and sufficient criterion for the identifiability of ( | , | ¯ ) in Markovian models called recanting
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witness criterion. The recanting witness criterion is satisfied when there is a variable along the causal path
connected to through another causal path not in . Consider an example causal model whose causal graph
is shown in Figure 8(d), when the causal path that we follow with interest = { → → → } with
as witness, then the recanting witness criterion is satisfied. The corresponding graph structure is called “kite”
graph. When this criterion is satisfied, ( | , | ¯ ) is not identifiable and ( ) is not identifiable also.
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Shpitser et al. [102] generalized this criterion to semi-Markovian models known as recanting district criterion.
Specifically, if there exists district that represents the set of variables not belonging to the set of sensitive
attributes , but ancestral of decision via a directed path which does not intersect , and nodes , ∈
(possibly = ), such that there is a causal path → → ... → in and a causal path → → ... →
not in , then the path-specific effect of on is not identifiable.

7.2.2. Efforts for dealing with identifiable issues
In Section 7.2.1, we show that the causal effects are not always identifiable only from observational data and
causal graphs. Several causality-based fairness methods have been proposed from different perspectives to
deal with identifiable issues.

Most previous approaches tend to make simplified or even unrealistic assumptions to avoid unidentifiable sit-
uations. For example, to avoid the unidentifiable issue of the counterfactual effect, Kusner et al. [11] adopted
three different assumptions: (i) only using non-descendants of the sensitive attributes to build the classifier;
(ii) postulating and inferring the non-deterministic sub-situations of the hidden variables based on domain
knowledge; and (iii) postulating the complete causal model, treating it as the additive noise model, and then
estimating the errors. Zhang et al. [26] evaded the unidentifiable issue of path-specific effect caused by satisfy-
ing recanting witness criterion via changing the causal model, i.e., cutting off all causal paths from sensitive
variables to the decision that pass through the redline variables. However, such simplified assumptions modify
the causal model equivalent to “redefining success”. Although these methods made simplified assumptions to
avoid identifiable issues, such assumptions may severely damage the performance of the decision model and
impose uncertainty on these methods. Besides, such simplified assumptions may modify the causal model
equivalent to “redefining success”, while any kind of repair is not expected within a modified model, which
results in fair inferences in the real world.

Recently, someworkarounds fordealing with unidentifiable situations aim tostay within the true causal model,
but they obtain the true unidentifiable causal effects by developing the upper and lower bounds of the causal
effects. For example, Wu et al. [57] mathematically developed the upper and lower bounds of counterfactual
fairness in unidentifiable situations and used a post-processing method for reconstructing trained classifiers
to make counterfactual fairness. Zhang et al. [27] mathematically bound indirect discrimination as the path-
specific effect in unidentifiable cases and proposed a pre-processing method for reconstructing the observa-
tional data to remove the discrimination from the original dataset. Hu et al. [108] adopted implicit generative
models and adversarial learning to estimate the upper and lower bound of average causal effect under uniden-
tifiable cases.

One of the major reasons causal effects are not identifiable is the presence of hidden confounding. Most
previous works [12,27,44,57] adopt the no hidden confounders assumption (i.e., Markovian model) to facilitate
the assessment of the causal effects. However, in practical scenarios, the existence of hidden confounders is
an inescapable fact, since measuring all possible confounders is impossible. For example, in many cases, we
cannot measure variables such as personal preferences, most genetic factors, and environmental factors. In
these cases, to deal with hidden confounders and identifiable issues, many studies adopt the potential outcome
framework [33,34] and are devoted to finding so-called “proxy variables” that reflect the information of hidden
confounders. For example, we cannot directly measure the socioeconomic status of patients, but patients’ de-