Computing the partition function, $Z$, of an Ising model over a graph of $N$ \enquote{spins} is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $\lambda$-fractional interpolation, $Z^{(\lambda)}$, where $\lambda=0$ and $\lambda=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)} \geq Z^{(\lambda)} \geq Z^{(BP)}$, and there exists a unique (\enquote{exact}) $\lambda_*$ such that $Z=Z^{(\lambda_*)}$. Generalizing the re-parametrization approach of \citep{wainwright_tree-based_2002} and the loop series approach of \citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall \lambda:\ Z=Z^{(\lambda)}{\tilde Z}^{(\lambda)}$, where the multiplicative correction, ${\tilde Z}^{(\lambda)}$, is an expectation over a node-independent probability distribution built from node-wise fractional marginals. Our theoretical analysis is complemented by extensive experiments with models from Ising ensembles over planar and random graphs of medium and large sizes. Our empirical study yields a number of interesting observations, such as the ability to estimate ${\tilde Z}^{(\lambda)}$ with $O(N^{2::4})$ fractional samples and suppression of variation in $\lambda_*$ estimates with an increase in $N$ for instances from a particular random Ising ensemble, where $[2::4]$ indicates a range from $2$ to $4$. We also discuss the applicability of this approach to the problem of image de-noising.

翻译：在包含N个“自旋”的图上计算伊辛模型的配分函数Z很可能是指数级依赖于N的。高效的变分方法，如信念传播(BP)和树重加权(TRW)算法，通过最小化相应的(BP或TRW)自由能来近似计算Z。我们通过构建一个λ-分数插值Z^{(\lambda)}来推广该变分方案，其中λ=0和λ=1分别对应TRW和BP近似。这一分数方案——称为分数信念传播(FBP)——保证了在吸引(铁磁)情况下Z^{(TRW)} ≥ Z^{(\lambda)} ≥ Z^{(BP)}，并且存在唯一的(“精确”)λ_*使得Z=Z^{(\lambda_*)}。通过推广\citep{wainwright_tree-based_2002}的重参数化方法和\citep{chertkov_loop_2006}的环级数方法，我们展示了如何将Z表示为乘积形式：∀ λ: Z=Z^{(\lambda)}{\tilde Z}^{(\lambda)}，其中乘法修正项{\tilde Z}^{(\lambda)}是基于节点分数边缘分布构建的节点独立概率分布上的期望值。我们的理论分析辅以对中等和大规模平面图及随机图上伊辛系综模型的广泛实验。我们的实证研究得出了若干有趣的观察结果，例如能够用O(N^{2::4})个分数样本估计{\tilde Z}^{(\lambda)}，以及对于来自特定随机伊辛系综的实例，λ_*估计值的方差随N增加而受到抑制，其中[2::4]表示从2到4的范围。我们还讨论了该方法在图像去噪问题中的适用性。