Adaptive Penalized Estimation of Directed Acyclic Graphs From Categorical Data
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We develop in this article a penalized likelihood method to estimate sparse Bayesian networks from categorical data. The structure of a Bayesian network is represented by a directed acyclic graph (DAG). We model the conditional distribution of a node given its parents by multi-logit regression and estimate the structure of a DAG via maximizing a regularized likelihood. The adaptive group Lasso penalty is employed to encourage sparsity by selecting grouped dummy variables encoding the level of a factor. We develop a blockwise coordinate descent algorithm to solve the penalized likelihood problem subject to the acyclicity constraint of a DAG. When intervention data are available, our method may construct a causal network, in which a directed edge represents a causal relation. We apply our method to various simulated networks and a real biological network. The results show that our method is very competitive, compared to other existing methods, in DAG estimation from both interventional and high-dimensional observational data. We also establish consistency in parameter and structure estimation for our method when the number of nodes is fixed.
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