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Information related measures are useful tools for multi variable dataanalysis, as measures of dependence among variables, and as descriptions oforder in biological and physical systems. Information related measures, likemarginal entropies, mutual / interaction / multi-information, have been used ina number of fields including descriptions of systems complexity and biologicaldata analysis. The mathematical relationships among these measures aretherefore of significant interest. Relations between common informationmeasures include the duality relations based on M\"obius inversion on lattices.These are the direct consequence of the symmetries of the lattices of the setsof variables (subsets ordered by inclusion). While the mathematical propertiesand relationships among these information-related measures are of significantinterest, there has been, to our knowledge, no systematic examination of thefull range of relationships and no unification of this diverse range offunctions into a single formalism as we do here. In this paper we defineoperators on functions on these lattices based on the M\"obius inversion ideathat map the functions into one another (M\"obius operators.) We show thatthese operators form a simple group isomorphic to the symmetric group S3.Relations among the set of functions on the lattice are transparently expressedin terms of the operator algebra, and, applied to the information measures, canbe used to derive a wide range of relationships among measures. We describe adirect relation between sums of conditional log-likelihoods and previouslydefined dependency measures. The algebra is naturally generalized which yieldsmore extensive relationships. This formalism provides a fundamental unificationof information related measures, but isomorphism of all distributive latticeswith the subset lattice implies broad potential application of these results.

Multivariate information measures: a unification using M\"obius operators on subset lattices