Toric ideals of characteristic imsets via quasi-independence gluing. (English) Zbl 07886937
Summary: Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper, we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gröbner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gröbner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and provide direction for future work.
MSC:
| 62Rxx | Statistics on algebraic and topological structures |
| 62E10 | Characterization and structure theory of statistical distributions |
| 62H22 | Probabilistic graphical models |
| 62D20 | Causal inference from observational studies |
| 62R01 | Algebraic statistics |
| 13P25 | Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
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