Distinguish Markov Equivalence Classes from Large-Scale Linear Non-Gaussian Data

In the problem of causal discovery, conditional independence (CI) tests are generally used to detect the causal relationships among observed data. Due to the curse of dimensionality and the limitation of causal direction learning based on
$V$
-structure learning, it is difficult for constraint-based methods to distinguish the actual graph from a set of Markov equivalence classes. To alleviate this problem, in this work, a novel regression-based method to test CIs over linear Non-Gaussian data is proposed. The main purpose of this proposal is to relax the CI test of
$x\bot y|Z$
to two unconditional independence tests
$x-f\left ({Z }\right) \bot y-g\left ({Z }\right) +\varSigma H\left ({Z }\right)$
and
$x-f\left ({Z }\right) +\varSigma H\left ({Z }\right) \bot y-g\left ({Z }\right)$
, where
$f$
and
$g$
can be estimated by linear regression independently. In addition, we further show that
$x-f\left ({Z }\right) \bot y-g\left ({Z }\right) +\varSigma H\left ({Z }\right)$
(or
$x-f\left ({Z }\right) +\varSigma H\left ({Z }\right) \bot y-g\left ({Z }\right)$
) can lead to
$x\gets Z$
(or
$y\gets Z$
). According to this regression-based method, we design a causal structure learning algorithm to learn the actual graph instead of a set of Markov equivalence classes over the observed data. Experiments indicate that our method can detect much more causal relationships than other existing methods, especially in large-scale cases.