Saturated Conditional Independence with Fixed and Undetermined Sets of Incomplete Random Variables

Abstract

The implication problem for saturated condi-tional independence statements is studied in the presence of fixed and undetermined sets of in-complete random variables. Here, random vari-ables are termed incomplete since they admit missing data. Two different notions of implica-tion arise. In the classic notion of V-implication, a statement is implied jointly by a set of state-ments and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of state-ments alone, leaving the set of random vari-ables undetermined. A first axiomatization for V-implication is established that distinguishes purely implied from V-implied statements. Ax-iomatic, algorithmic and logical characteriza-tions of pure implication are established. Pure implication appeals to applications in which the existence of random variables is uncertain, for example, when independence statements are in-tegrated from different sources, when random variables are unknown or shall remain hidden. 1