I am reading Judea Pearl's "Causality" (second edition 2009) and in section 1.1.5 Conditional Independence and Graphoids, he states:
The following is a (partial) list of properties satisfied by the conditional independence relation (X_||_Y | Z).
- Symmetry: (X_||_ Y | Z) ==> (Y_||_X | Z).
- Decomposition: (X_||_ YW | Z) ==> (X_||_Y | Z).
- Weak union: (X_||_ YW | Z) ==> (X_||_Y | ZW).
- Contraction: (X_||_ Y | Z) & (X_||_ W | ZY) ==> (X_||_ YW | Z).
- Intersection: (X_||_ W | ZY) & (X_||_ Y | ZW) (X_||_ YW | Z).
(Intersection is valid in strictly positive probability distributions.)
(formula (1.28) given earlier in the publicatiob: [(X_||_ Y | Z) iff P(X | Y,Z ) = P(X | Z) )
But what is an "strictly positive distribution" in general terms, and what distinguishes a "strictly positive distribution" form a distribution that is not strictly positive?