A note on the relationship between conditional and unconditional independence and its extensions for Markov kernels

Abstract

Two known results 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 to obtain a minimal condition, which, added to conditional independence, implies independence. Some examples, counterexamples, and representation results are provided to clarify the concepts introduced and the propositions of the statement of the main theorem. Moreover, conditional independence and the mentioned results are extended to the framework of Markov kernels.