Alternative Markov and Causal Properties for Acyclic Directed Mixed Graphs
This work addresses a theoretical problem in graphical models for statisticians and AI researchers, offering incremental extensions to existing frameworks.
The paper tackles the problem of extending Andersson-Madigan-Perlman chain graphs by relaxing constraints to allow directed cycles and multiple edges, introducing new Markov properties and showing their equivalence for positive distributions. It results in an interpretation as structural equation systems with correlated errors, enabling adaptation of Pearl's do-calculus and an exact learning algorithm via answer set programming.
We extend Andersson-Madigan-Perlman chain graphs by (i) relaxing the semidirected acyclity constraint so that only directed cycles are forbidden, and (ii) allowing up to two edges between any pair of nodes. We introduce global, and ordered local and pairwise Markov properties for the new models. We show the equivalence of these properties for strictly positive probability distributions. We also show that when the random variables are continuous, the new models can be interpreted as systems of structural equations with correlated errors. This enables us to adapt Pearl's do-calculus to them. Finally, we describe an exact algorithm for learning the new models from observational and interventional data via answer set programming.