Identifiability of non-isolated mixed ordinal–exponential-family edges

Determine whether distributional identifiability is restored for a mixed ordinal–exponential-family edge whose endpoints have additional neighboring nodes, in the parameter regimes where the exponential-family node has two support points with an affine sufficient statistic and canonical link or the ordinal node has two categories with an affine sufficient statistic.

Background

The multivariate converse proves non-identifiability for an ordinal–exponential-family edge under the limiting support or category conditions only when the two endpoints form an isolated two-node connected component of the skeleton. The construction relies on embedding the bivariate non-identifiable models without additional constraints from neighboring nodes.

For edges that are not isolated, the other parents and children may impose compatibility constraints that invalidate the bivariate construction. The paper explicitly leaves unresolved whether such additional graph structure can restore identifiability.

References

When the edge is not isolated, the remaining parents of the endpoints impose constraints that the bivariate construction need not satisfy, and whether identifiability is restored at such an edge is beyond the scope of this paper.

— On the Identifiability of Mixed Ordinal and Exponential Family Causal DAGs under Linear Parametric Models  (2609.17942 - Mishra et al., 16 Sep 2026) in Section 3.3, final paragraph of the Multivariate Identifiability section