Properties of Bethe Free Energies and Message Passing in Gaussian Models
Properties of Bethe Free Energies and Message Passing in Gaussian Models
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高斯模型中 Bethe 自由能的性质和消息传递
DOI：
10.1613/jair.3195
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
T. Heskes
中科院分区：
文献类型：
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作者：
Botond Cseke;T. Heskes
我们用平均场和分数贝特近似解决了高斯概率模型中近似边际的计算问题。我们用近似边缘的力矩参数定义了高斯分数阶贝特自由能，导出了分数阶贝特自由能的下界和上界，并建立了下界由下有界的必要条件。结果表明，该条件与消息传递算法收敛的充分条件——两两归一化条件是相同的。我们证明了高斯消息传递算法的稳定不动点是高斯贝特自由能的局部极小值。通过一个反例，证明了自由能的无界性意味着消息传递算法的发散性的猜想。
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Bethe free energy and establish a necessary condition for the lower bound to be bounded from below. It turns out that the condition is identical to the pairwise normalizability condition, which is known to be a sufficient condition for the convergence of the message passing algorithm. We show that stable fixed points of the Gaussian message passing algorithm are local minima of the Gaussian Bethe free energy. By a counterexample, we disprove the conjecture stating that the unboundedness of the free energy implies the divergence of the message passing algorithm.