A new result on the relationship between conditional and unconditional independence
Two known results (Phillips (1988) and Dawid (1979)) on the relationship between conditional and unconditional independence are obtained as a consequence of the main result of this paper, a theorem that uses independence of Markov kernels (see Nogales (2013)) to obtain a minimal condition which added to conditional independence implies independence. Namely, the mentioned result, part of a paper by Nogales and Pérez (2018, arXiv:1706.03955), reads as follows: Let X, Y, Z be three random variables. If X and Y are conditionally independent given Z, then X and Y are independent if and only if the conditional distributions of X and Y given Z are independent. This last condition is equivalent to the uncorrelatedness of the conditional expectations given Z of every pair of bounded real random functions of X and Y.
Some counterexamples and representation results are provided to clarify the concepts introduced and the propositions of the statement of the main theorem.
Palabras clave / Keywords: conditional independence Markov kernel
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