Identification of causal relations among variables is central to many scientific investigations, as in regulatory network analysis of gene interactions and brain network analysis of effective connectivity of causal relations between regions of interest. Statistically, causal relations are often modeled by a directed acyclic graph (DAG), and hence that reconstruction of a DAG’s structure leads to the discovery of causal relations. Yet, reconstruction of a DAG’s structure from observational data is impossible because a DAG Gaussian model is usually not identifiable with unequal error variances. In this article, we reconstruct a DAG’s structure with the help of interventional data. Particularly, we construct a constrained likelihood to regularize intervention in addition to adjacency matrices to identify a DAG’s structure, subject to an error variance constraint to further reinforce the model identifiability. Theoretically, we show that the proposed constrained likelihood leads to identifiable models, thus correct reconstruction of a DAG’s structure through parameter estimation even with unequal error variances. Computationally, we design efficient algorithms for the proposed method. In simulations, we show that the proposed method enables to produce a higher accuracy of reconstruction with the help of interventional observations.
Bibliographical noteFunding Information:
∗The authors thank the editor, the associate editor and anonymous referees for helpful comments and suggestions. Research supported in part by NSF grants DMS-1712564, DMS-1721216, DMS-1952539, and NIH grants 1R01GM126002, 2R01HL105397, 1R01AG065636, R01AG069895.
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- Causal relations
- Constrained likelihood
- Reconstruction identifiability