  Missing data are an unavoidable complication frequently encountered in many
causal discovery tasks. When a missing process depends on the missing values
themselves (known as self-masking missingness), the recovery of the joint
distribution becomes unattainable, and detecting the presence of such
self-masking missingness remains a perplexing challenge. Consequently, due to
the inability to reconstruct the original distribution and to discern the
underlying missingness mechanism, simply applying existing causal discovery
methods would lead to wrong conclusions. In this work, we found that the recent
advances additive noise model has the potential for learning causal structure
under the existence of the self-masking missingness. With this observation, we
aim to investigate the identification problem of learning causal structure from
missing data under an additive noise model with different missingness
mechanisms, where the `no self-masking missingness' assumption can be
eliminated appropriately. Specifically, we first elegantly extend the scope of
identifiability of causal skeleton to the case with weak self-masking
missingness (i.e., no other variable could be the cause of self-masking
indicators except itself). We further provide the sufficient and necessary
identification conditions of the causal direction under additive noise model
and show that the causal structure can be identified up to an IN-equivalent
pattern. We finally propose a practical algorithm based on the above
theoretical results on learning the causal skeleton and causal direction.
Extensive experiments on synthetic and real data demonstrate the efficiency and
effectiveness of the proposed algorithms.
