  This paper considers an extension of the linear non-Gaussian acyclic model
(LiNGAM) that determines the causal order among variables from a dataset when
the variables are expressed by a set of linear equations, including noise. In
particular, we assume that the variables are binary. The existing LiNGAM
assumes that no confounding is present, which is restrictive in practice. Based
on the concept of independent component analysis (ICA), this paper proposes an
extended framework in which the mutual information among the noises is
minimized. Another significant contribution is to reduce the realization of the
shortest path problem. The distance between each pair of nodes expresses an
associated mutual information value, and the path with the minimum sum (KL
divergence) is sought. Although $p!$ mutual information values should be
compared, this paper dramatically reduces the computation when no confounding
is present. The proposed algorithm finds the globally optimal solution, while
the existing locally greedily seek the order based on hypothesis testing. We
use the best estimator in the sense of Bayes/MDL that correctly detects
independence for mutual information estimation. Experiments using artificial
and actual data show that the proposed version of LiNGAM achieves significantly
better performance, particularly when confounding is present.
