Approximate Reasoning about Generalized Conditional Independence with Complete Random Variables

The implication problem of conditional statements about the independence of finitely many sets of random variables is studied in the presence of controlled uncertainty. Uncertainty refers to the possibility of missing data. As a control mechanism random variables can be declared complete, in which case data on these random variables cannot be missing. While the implication of conditional independence statements is not axiomatizable, a finite Horn axiomatization is established for the expressive class of saturated conditional independence statements under controlled uncertainty. Complete random variables allow us to balance the expressivity of sets of saturated statements with the efficiency of deciding their implication. This ability can soundly approximate reasoning in the absence of missing data. Delobel’s class of full first-order hierarchical database decompositions are generalized to the presence of controlled uncertainty, and their implication problem shown to be equivalent to that of saturated conditional independence.

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