16,737 research outputs found
Loop Calculus in Statistical Physics and Information Science
Considering a discrete and finite statistical model of a general position we
introduce an exact expression for the partition function in terms of a finite
series. The leading term in the series is the Bethe-Peierls (Belief
Propagation)-BP contribution, the rest are expressed as loop-contributions on
the factor graph and calculated directly using the BP solution. The series
unveils a small parameter that often makes the BP approximation so successful.
Applications of the loop calculus in statistical physics and information
science are discussed.Comment: 4 pages, submitted to Phys.Rev.Lett. Changes: More general model,
Simpler derivatio
Loop Calculus for Non-Binary Alphabets using Concepts from Information Geometry
The Bethe approximation is a well-known approximation of the partition
function used in statistical physics. Recently, an equality relating the
partition function and its Bethe approximation was obtained for graphical
models with binary variables by Chertkov and Chernyak. In this equality, the
multiplicative error in the Bethe approximation is represented as a weighted
sum over all generalized loops in the graphical model. In this paper, the
equality is generalized to graphical models with non-binary alphabet using
concepts from information geometry.Comment: 18 pages, 4 figures, submitted to IEEE Trans. Inf. Theor
Loop series for discrete statistical models on graphs
In this paper we present derivation details, logic, and motivation for the
loop calculus introduced in \cite{06CCa}. Generating functions for three
inter-related discrete statistical models are each expressed in terms of a
finite series. The first term in the series corresponds to the Bethe-Peierls
(Belief Propagation)-BP contribution, the other terms are labeled by loops on
the factor graph. All loop contributions are simple rational functions of spin
correlation functions calculated within the BP approach. We discuss two
alternative derivations of the loop series. One approach implements a set of
local auxiliary integrations over continuous fields with the BP contribution
corresponding to an integrand saddle-point value. The integrals are replaced by
sums in the complimentary approach, briefly explained in \cite{06CCa}. A local
gauge symmetry transformation that clarifies an important invariant feature of
the BP solution, is revealed in both approaches. The partition function remains
invariant while individual terms change under the gauge transformation. The
requirement for all individual terms to be non-zero only for closed loops in
the factor graph (as opposed to paths with loose ends) is equivalent to fixing
the first term in the series to be exactly equal to the BP contribution.
Further applications of the loop calculus to problems in statistical physics,
computer and information sciences are discussed.Comment: 20 pages, 3 figure
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