D. Geiger et J. Pearl, LOGICAL AND ALGORITHMIC PROPERTIES OF CONDITIONAL-INDEPENDENCE AND GRAPHICAL MODELS, Annals of statistics, 21(4), 1993, pp. 2001-2021
This article develops an axiomatic basis for the relationship between
conditional independence and graphical models in statistical analysis.
In particular, the following relationships are established: (1) every
axiom for conditional independence is an axiom for graph separation,
(2) every graph represents a consistent set of independence and depend
ence constraints, (3) all binary factorizations of strictly positive p
robability models can be encoded and determined in polynomial time usi
ng their correspondence to graph separation, (4) binary factorizations
of non-strictly positive probability models can also be derived in po
lynomial time albeit less efficiently and (5) unconditional independen
ce relative to normal models can be axiomatized with a finite set of a
xioms.