Local Markov Property for Models Satisfying Composition Axiom
This work addresses a computational bottleneck in statistical modeling for researchers dealing with latent variables, though it appears incremental as it builds on existing axioms and graph theory.
The paper tackles the problem of reducing the exponential number of conditional independencies required for the local Markov property in DAGs with latent variables by showing that if probability distributions satisfy the composition axiom, the number can be reduced to linear in certain graph types, with applications in testing linear structural equation models with correlated errors.
The local Markov condition for a DAG to be an independence map of a probability distribution is well known. For DAGs with latent variables, represented as bi-directed edges in the graph, the local Markov property may invoke exponential number of conditional independencies. This paper shows that the number of conditional independence relations required may be reduced if the probability distributions satisfy the composition axiom. In certain types of graphs, only linear number of conditional independencies are required. The result has applications in testing linear structural equation models with correlated errors.