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Inference efforts -- required to compute partition function, $Z$, of an Isingmodel over a graph of $N$ ``spins" -- are most likely exponential in $N$.Efficient variational methods, such as Belief Propagation (BP) and TreeRe-Weighted (TRW) algorithms, compute $Z$ approximately minimizing respective(BP- or TRW-) free energy. We generalize the variational scheme building a$\lambda$-fractional-homotopy, $Z^{(\lambda)}$, where $\lambda=0$ and$\lambda=1$ correspond to TRW- and BP-approximations, respectively, and$Z^{(\lambda)}$ decreases with $\lambda$ monotonically. Moreover, thisfractional scheme guarantees that in the attractive (ferromagnetic) case$Z^{(TRW)}\geq Z^{(\lambda)}\geq Z^{(BP)}$, and there exists a unique(``exact") $\lambda_*$ such that, $Z=Z^{(\lambda_*)}$. Generalizing there-parametrization approach of \cite{wainwright_tree-based_2002} and the loopseries approach of \cite{chertkov_loop_2006}, we show how to express $Z$ as aproduct, $\forall \lambda:\ Z=Z^{(\lambda)}{\cal Z}^{(\lambda)}$, where themultiplicative correction, ${\cal Z}^{(\lambda)}$, is an expectation over anode-independent probability distribution built from node-wise fractionalmarginals. Our theoretical analysis is complemented by extensive experimentswith models from Ising ensembles over planar and random graphs of medium- andlarge- sizes. The empirical study yields a number of interesting observations,such as (a) ability to estimate ${\cal Z}^{(\lambda)}$ with $O(N^4)$ fractionalsamples; (b) suppression of $\lambda_*$ fluctuations with increase in $N$ forinstances from a particular random Ising ensemble.

Exact Fractional Inference via Re-Parametrization & Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms