Bayesian networks are of paramount significance in modeling joint probability distributions and have garnered extensive applications across diverse domains. The continuous optimization method formulates the structure learning problem as a purely continuous optimization problem within the space of real matrices, thereby offering a novel avenue for learning directed acyclic graphs (DAGs). In our quest to enhance performance and interpretability, we introduce a groundbreaking continuous optimization approach for learning the structures of DAGs, namely DAGs structure learning with Absorbing Markov Chain (DAG-AMC). DAG-AMC ingeniously transforms the acyclic constraint into a node transition challenge, effectively recasting it as a Markov chain problem with absorbing states. This innovative transformation reconceptualizes the graph structure as a state transition matrix within the framework of an absorbing Markov chain. The absorption time intrinsic to this chain provides an elegant representation of the acyclic constraint in DAG structure learning. We leverage the augmented Lagrangian method, incorporating the constructed smooth function as a constraint throughout the optimization process. Empirical experiments conducted on both synthetic and real-world datasets highlight the remarkable efficacy of our proposed DAG-AMC. Our results consistently surpass those of baseline methods across a wide array of evaluation metrics, thus underscoring the superior potential of DAG-AMC as a preeminent solution for DAG structure learning.

Discovering causal relationships among observed variables is an important research focus in data mining. Existing causal discovery approaches are mainly based on constraint-based methods and functional causal models (FCMs). However, the constraint-based method cannot identify the Markov equivalence class and the functional causal models cannot identify the complex interrelationships when multiple variables affect one variable. To address the two aforementioned problems, we propose a new graph structure Causal Star Graph (CSG) and a corresponding framework Causal Discovery via Causal Star Graphs (CD-CSG) to divide a causal directed acyclic graph into multiple CSGs for causal discovery. In this framework, we also propose a generalized learning in CSGs based on a variational approach to learn the representative intermediate variable of CSG’s non-central variables. Through the generalized learning in CSGs, the asymmetry in the forward and backward model of CD-CSG can be found to identify the causal directions in the directed acyclic graphs. We further divide the CSGs into three categories and provide the causal identification principle under each category in our proposed framework. Experiments using synthetic data show that the causal relationships between variables can be effectively identified with CD-CSG and the accuracy of CD-CSG is higher than the best existing model. By applying CD-CSG to real-world data, our proposed method can greatly augment the applicability and effectiveness of causal discovery.

Discovering causal relationships among observed variables is a new research focus in the area of data mining. Methods based on the additive noise model have been proved to be efficient in the identification of cause-effect pairs. However, when trying to determine many-to-one causality, additive noise models often fail to identify the causal direction due to the complex interrelationships and interactions even though the generation of each causal relation follows the additive noise model, and become unreliable in practical applications. In this work, to identify the causal direction, we propose a Hierarchical Additive Noise Model (HANM) to convert many-to-one causality into an approximate one-to-one causality by generalizing multiple factors into an intermediate variable with a variational approach, and use asymmetry in the forward model and backward model of HANM to identify causal direction. Experiments using synthetic data show that many-to-one causality can be effectively identified through asymmetry with our proposed HANM and the accuracy of HANM is higher than the best existing model. By applying the model to real-world data, it can be seen that HANM can greatly augment the application scope of functional causal models for causal discovery.