# Penalized Estimation of Directed Acyclic Graphs From Discrete Data

arXiv: https://arxiv.org/abs/1403.2310 · HTML: https://mlsanity.com/paper/1403.2310/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Mar 2014.

## Built on by (most cited first, 10 of 10)

- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Simultaneous Missing Value Imputation and Structure Learning with Groups](https://mlsanity.com/paper/2110.08223/) (2021)
- [Distributed Learning of Generalized Linear Causal Networks](https://mlsanity.com/paper/2201.09194/) (2022)
- [Recursive Causal Discovery](https://mlsanity.com/paper/2403.09300/) (2024)
- [Markov Equivalence and Consistency in Differentiable Structure Learning](https://mlsanity.com/paper/2410.06163/) (2024)
- [Partitioned hybrid learning of Bayesian network structures](https://mlsanity.com/paper/2103.12188/) (2021)
- [A Stochastic Variance-Reduced Coordinate Descent Algorithm for Learning Sparse Bayesian Network from Discrete High-Dimensional Data](https://mlsanity.com/paper/2108.09501/) (2021)
- [Causal Discovery on Dependent Binary Data](https://mlsanity.com/paper/2412.20289/) (2024)
- [Differentiable Structure Learning and Causal Discovery for General Binary Data](https://mlsanity.com/paper/2509.21658/) (2025)

## Abstract (arXiv)

Bayesian networks, with structure given by a directed acyclic graph (DAG), are a popular class of graphical models. However, learning Bayesian networks from discrete or categorical data is particularly challenging, due to the large parameter space and the difficulty in searching for a sparse structure. In this article, we develop a maximum penalized likelihood method to tackle this problem. Instead of the commonly used multinomial distribution, we model the conditional distribution of a node given its parents by multi-logit regression, in which an edge is parameterized by a set of coefficient vectors with dummy variables encoding the levels of a node. To obtain a sparse DAG, a group norm penalty is employed, and a blockwise coordinate descent algorithm is developed to maximize the penalized likelihood subject to the acyclicity constraint of a DAG. When interventional data are available, our method constructs a causal network, in which a directed edge represents a causal relation. We apply our method to various simulated and real data sets. The results show that our method is very competitive, compared to many existing methods, in DAG estimation from both interventional and high-dimensional observational data.