Joint response graphs and separation induced by triangular systems

Research Report SRR01-002

Joint response graphs and separation induced by triangular systems

Nanny Wermuth and D.R. Cox

Abstract: Probability distributions generated over a directed acyclic graph are considered here as what we call triangular systems. The graph captures the independence structure of the system. In econometrics, linear triangular systems have been called univariate recursive equations with uncorrelated residuals. Separation results provide criteria for deciding whether any chosen conditional independence statement is implied by a given independence graph. We state and prove separation in triangular systems as block-diagonality of a part of the edge matrix of the generating graph, having been transformed in a particular way. For the proof we derive a matrix result for orthogonalising weighted sums of variables. We apply the result also to orthogonalise blocks of vector variables defined for linear triangular systems to obtain corresponding edge matrices of what we call joint response graphs. The results for transforming the edge matrix in a linear triangular system hold for all Gaussian distributions generated over the given directed acyclic graph and imply factorizations of densities in marginal and conditional distributions. As a consequence of such factorization properties the results are shown to apply in unchanged form to any joint probability distribution generated over the same directed acyclic graph. There is thus an essential equivalence between various matrix transformations and graphical properties. Use of the former may be considerably more direct for complex graphs.

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