Expectation Propagation (EP) is a widely used iterative message-passing algorithm that decomposes a global inference problem into multiple local ones. It approximates marginal distributions as ``beliefs'' using intermediate functions called ``messages''. It has been shown that the stationary points of EP are the same as corresponding constrained Bethe Free Energy (BFE) optimization problem. Therefore, EP is an iterative method of optimizing the constrained BFE. However, the iterative method may fall out of the feasible set of the BFE optimization problem, i.e., the beliefs are not integrable. In most literature, the authors use various methods to keep all the messages integrable. In most Bayesian estimation problems, limiting the messages to be integrable shrinks the actual feasible set. Furthermore, in extreme cases where the factors are not integrable, making the message itself integrable is not enough to have integrable beliefs. In this paper, two EP frameworks are proposed to ensure that EP has integrable beliefs. Both of the methods allows non-integrable messages. We then investigate the signal recovery problem in Generalized Linear Model (GLM) using our proposed methods.

翻译：期望传播（EP）是一种广泛使用的迭代消息传递算法，它将全局推理问题分解为多个局部问题。该算法通过称为"消息"的中间函数，将边缘分布近似为"信念"。已有研究表明，EP的驻点与对应的约束Bethe自由能（BFE）优化问题一致。因此，EP是优化约束BFE的一种迭代方法。然而，该迭代方法可能偏离BFE优化问题的可行集，即信念不可积。在现有文献中，作者通常采用各种方法确保所有消息可积。在大多数贝叶斯估计问题中，限制消息可积会缩小实际可行集。此外，在因子不可积的极端情况下，仅使消息本身可积并不足以获得可积的信念。本文提出了两种EP框架以确保EP具有可积信念，这两种方法均允许不可积消息的存在。随后，我们采用所提方法研究了广义线性模型（GLM）中的信号恢复问题。