  We consider the problem of inferring the causal structure from observational
data, especially when the structure is sparse. This type of problem is usually
formulated as an inference of a directed acyclic graph (DAG) model. The linear
non-Gaussian acyclic model (LiNGAM) is one of the most successful DAG models,
and various estimation methods have been developed. However, existing methods
are not efficient for some reasons: (i) the sparse structure is not always
incorporated in causal order estimation, and (ii) the whole information of the
data is not used in parameter estimation. To address {these issues}, we propose
a new estimation method for a linear DAG model with non-Gaussian noises. The
proposed method is based on the log-likelihood of independent component
analysis (ICA) with two penalty terms related to the sparsity and the
consistency condition. The proposed method enables us to estimate the causal
order and the parameters simultaneously. For stable and efficient optimization,
we propose some devices, such as a modified natural gradient. Numerical
experiments show that the proposed method outperforms existing methods,
including LiNGAM and NOTEARS.
