Markov Properties for Linear Causal Models with Correlated Errors

Abstract

A linear causal model with correlated errors, represented by a DAG with bi-directed edges, can be tested by the set of conditional independence relations implied by the model. A global Markov property specifies, by the d-separation criterion, the set of all conditional independence relations holding in any model associated with a graph. A local Markov property specifies a much smaller set of conditional independence relations which will imply all other conditional independence relations which hold under the global Markov property. For DAGs with bi-directed edges associated with arbitrary probability distributions, a local Markov property is given in Richardson (2003) which may invoke an exponential number of conditional independencies. In this paper, we show that for a class of linear structural equation models with correlated errors, there is a local Markov property which will invoke only a linear number of conditional independence relations. For general linear models, we provide a local Markov property that often invokes far fewer conditional independencies than that in Richardson (2003). The results have applications in testing linear structural equation models with correlated errors.

Index Terms

• Computing methodologies

  • Machine learning

    • Machine learning approaches

      • Factorization methods

        • Canonical correlation analysis

  • Modeling and simulation

    • Model development and analysis

      • Model verification and validation

      • Modeling methodologies

• Mathematics of computing

  • Probability and statistics

    • Probabilistic representations

      • Markov networks

    • Statistical paradigms

      • Regression analysis

    • Stochastic processes

      • Markov processes

• Theory of computation

  • Theory and algorithms for application domains

    • Machine learning theory

      • Markov decision processes