Factorizing Probabilistic Graphical Models Using Cooccurrence Rate
Clinical Orthopaedics and Related Research(2011)
摘要
Factorization is of fundamental importance in the area of probabilistic
graphical models (PGMs). In this paper, we theoretically develop the novel
mathematical concept \textbf{Cooccurrence Rate (CR)} for factorizing PGMs.
Comparing with the existing methods, our method has three obvious advantages:
(1) CR provides a unified mathematical foundation for factorizing different
types of PGMs, no matter they are directed or undirected, cyclic or acyclic;
(2) comparing with clique potentials in MRF factorizations, CR has clear
probability definitions and intuitive interpretations; (3) using CR to
factorize a PGM, each factorizing step corresponds to a bi-partition operation
or a merge operation on the graph, which allows us to leverage graph
partitioning or graph clustering techniques to factorized PGMs.
We further describe three ways to learn CR values, which brings CR to be
practically useful. Using CR to factorize PGMs, we should explicitly respect
both of the independent or conditionally independent (ICI) semantics and the
local probability constraint (LPC) semantics implied by the representation of
PGMs.
We also propose approximate techniques to further simplify CR factorizations.
Analyses on the relationships between the CR and MRF factorizations are also
conducted. The results show that: (1) for TCG graphs, the clique potentials in
MRF factorizations can be expressed as CR functions. This means that CR
provides clear and meaningful probability interpretations for the clique
potentials; (2) for CCG graphs, clique potentials can not be expressed as CR
functions. This means that from the view of CR, MRF factorizations are not
exact factorizations for CCG graphs. But MRF factorizations can be
approximations of the exact CR factorizations.
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关键词
conditional independence,artificial intelligent,bayesian network,conditional random field,graph partitioning,graph clustering
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