Logics for Approximating Implication Problems of Saturated Conditional Independence

Random variables are declared complete whenever they must not admit missing data. Intuitively, the larger the set of complete random variables the closer the implication of saturated conditional independence statements is approximated. Two different notions of implication are studied. In the classical notion, a statement is implied jointly by a set of statements, the fixed set of random variables and its subset of complete random variables. For the notion of pure implication the set of random variables is left undetermined. A first axiomatization for the classical notion is established that distinguishes purely implied from classically implied statements. Axiomatic, algorithmic and logical characterizations of pure implication are established. The latter appeal to applications in which the existence of random variables is uncertain, for example, when statements are integrated from different sources, when random variables are unknown or when they shall remain hidden.

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