Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models
This work allows for more flexible causal discovery in latent variable models, though it is an incremental extension of existing theory.
The paper extends the Trek Separation Theorem to apply to partially non-linear and cyclic models, enabling constraint-based causal search algorithms to discover causal structures in a broader class of latent variable models.
The Trek Separation Theorem (Sullivant et al. 2010) states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. This justifies application of constraint-based causal search algorithms such as the BuildPureClusters algorithm (Silva et al. 2006) for discovering the causal structure of latent variable models to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables.