Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multivariate information measures: a unification using M\"obius
  operators on subset lattices   Information related measures are useful tools for multi variable data
analysis, as measures of dependence among variables, and as descriptions of
order in biological and physical systems. Information related measures, like
marginal entropies, mutual / interaction / multi-information, have been used in
a number of fields including descriptions of systems complexity and biological
data analysis. The mathematical relationships among these measures are
therefore of significant interest. Relations between common information
measures include the duality relations based on M\"obius inversion on lattices.
These are the direct consequence of the symmetries of the lattices of the sets
of variables (subsets ordered by inclusion). While the mathematical properties
and relationships among these information-related measures are of significant
interest, there has been, to our knowledge, no systematic examination of the
full range of relationships and no unification of this diverse range of
functions into a single formalism as we do here. In this paper we define
operators on functions on these lattices based on the M\"obius inversion idea
that map the functions into one another (M\"obius operators.) We show that
these operators form a simple group isomorphic to the symmetric group S3.
Relations among the set of functions on the lattice are transparently expressed
in terms of the operator algebra, and, applied to the information measures, can
be used to derive a wide range of relationships among measures. We describe a
direct relation between sums of conditional log-likelihoods and previously
defined dependency measures. The algebra is naturally generalized which yields
more extensive relationships. This formalism provides a fundamental unification
of information related measures, but isomorphism of all distributive lattices
with the subset lattice implies broad potential application of these results.

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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: stat