Identifiability assumptions for directed graphical models with feedback

Directed graphical models provide a useful framework for modeling causal or directional relationships for multivariate data. Prior work has largely focused on identifiability and search algorithms for directed acyclic graphical (DAG) models. In many applications, feedback naturally arises and directed graphical models that permit cycles arise. However theory and methodology for directed graphical models with feedback are considerably less developed since graphs with cycles pose a number of additional challenges. In this paper we address the issue of identifiability for general directed cyclic graphical (DCG) models satisfying only the Markov assumption. In particular, in addition to the faithfulness assumption which has already been introduced for cyclic models, we introduce two new identifiability assumptions, one based on selecting the model with the fewest edges and the other based on selecting the DCG model that entails the maximum d-separation rules. We provide theoretical results comparing these assumptions which shows that: (1) selecting models with the largest number of d-separation rules is strictly weaker than the faithfulness assumption; (2) unlike for DAG models, selecting models with the fewest edges do not necessarily result in a milder assumption than the faithfulness assumption. We also provide connections between our two new principles and minimality assumptions which lead to a ranking of how strong and weak various identifiability and minimality assumptions are for both DAG and DCG models. We use our identifiability assumptions to develop search algorithms for small-scale DCG models. Our simulations results using our search algorithms support our theoretical results, showing that our two new principles generally out-perform the faithfulness assumption in terms of selecting the true skeleton for DCG models.