Local Markov Property for Models Satisfying Composition Axiom

TL;DR

This paper demonstrates that for models satisfying the composition axiom, the number of conditional independence relations needed for the local Markov property can be significantly reduced, especially in certain graph types, aiding in testing structural equation models.

Contribution

It establishes a reduced set of conditional independencies for models with latent variables satisfying the composition axiom, improving efficiency in model testing.

Findings

Number of required independencies can be linear in certain graphs.

Models satisfying the composition axiom need fewer conditional independencies.

Application to testing linear structural equation models with correlated errors.

Abstract

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.