Bayesian Approach to Linear Bayesian Networks
Seyong Hwang Kyoungjae Lee Sunmin Oh Gunwoong Park
This study proposes the first Bayesian approach for learning high-dimensional
linear Bayesian networks. The proposed approach iteratively estimates each
element of the topological ordering from backward and its parent using the
inverse of a partial covariance matrix. The proposed method successfully
recovers the underlying structure when Bayesian regularization for the inverse
covariance matrix with unequal shrinkage is applied. Specifically, it shows
that the number of samples $n = \Omega( d_M^2 \log p)$ and $n = \Omega(d_M^2
p^{2/m})$ are sufficient for the proposed algorithm to learn linear Bayesian
networks with sub-Gaussian and 4m-th bounded-moment error distributions,
respectively, where $p$ is the number of nodes and $d_M$ is the maximum degree
of the moralized graph. The theoretical findings are supported by extensive
simulation studies including real data analysis. Furthermore the proposed
method is demonstrated to outperform state-of-the-art frequentist approaches,
such as the BHLSM, LISTEN, and TD algorithms in synthetic data.