Bayesian Approach to Linear Bayesian Networks
This provides a novel Bayesian method for structure learning in high-dimensional Bayesian networks, addressing a known bottleneck in causal inference and graphical modeling.
This paper tackles the problem of learning high-dimensional linear Bayesian networks by proposing the first Bayesian approach, which successfully recovers the underlying structure with theoretical guarantees requiring sample sizes of n = Ω(d_M^2 log p) for sub-Gaussian errors and n = Ω(d_M^2 p^{2/m}) for bounded-moment errors, and outperforms state-of-the-art frequentist methods in simulations.
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 $n = Ω( d_M^2 \log p)$ and $n = Ω(d_M^2 p^{2/m})$ 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.