Markov Properties for Linear Causal Models with Correlated Errors

I finished a paper (with Jin Tian) entitled "Markov Properties for Linear Causal Models with Correlated Errors". This paper introduces an efficient way to test linear causal models (or linear structural equation models) with correlated errors using conditional independence tests. A linear causal model imposes a set of conditional independence constraints on the associated probability distribution, which can be read by the d-separation criterion. However, the set of conditional independence constraints is huge and the question arises whether there is a smaller set of conditional independence constraints that can derive all the others (this smaller set is called a basis in some literature).

This problem is equivalent to finding a local Markov property for models satisfying the composition axiom. A local Markov property for arbitrary models was given by Richardson (2003) which may invoke an exponential number of constraints. We show that for a class of linear causal models with correlated errors there is a local Markov property that invokes only linear number of constraints. For general models, we provide a local Markov property that often invokes far fewer number of constraints than that in Richardson (2003). This paper generalizes the result of Pearl (2000) which gives a local Markov property for linear causal models without correlated errors. A local Markov property for linear causal models with correlated errors was studied by Shipley (2003). The method in Shipley (2003) may or may not, depending on the models, be able to find a basis while our method always produces a basis.

This paper extends our previous work presented in UAI 2005 in several ways. It includes a few more local Markov properties that lead to more efficient tests. Also, it gives an improved method that can reduce the number of constraints for general linear causal models.