Gaussian Mixture Models (GMMs) commonly arise in communication systems, particularly in bilinear joint estimation and detection problems. Although the product of GMMs is still a GMM, as the number of factors increases, the number of components in the resulting product GMM grows exponentially. To obtain a tractable approximation for a univariate factored probability density function (PDF), such as a product of GMMs, we investigate iterative message-passing algorithms. Based on Belief Propagation (BP), we propose a Variable Duplication and Gaussian Belief Propagation (VDBP)-based algorithm. The key idea of VDBP is to construct a multivariate measurement model whose marginal posterior is equal to the given univariate factored PDF. We then apply Gaussian BP (GaBP) to transform the global inference problem into local ones. Expectation propagation (EP) is another branch of message passing algorithms. In addition to converting the global approximation problem into local ones, it features a projection operation that ensures the intermediate functions (messages) belong to a desired family. Due to this projection, EP can be used to approximate the factored PDF directly. However, even if every factor is integrable, the division operation in EP may still cause the algorithm to fail when the mean and variance of a non-integrable belief are required. Therefore, this paper proposes two methods that combine EP with our previously proposed techniques for handling non-integrable beliefs to approximate univariate factored distributions.

翻译：高斯混合模型（Gaussian Mixture Models, GMMs）在通信系统中普遍存在，尤其是在双线性联合估计与检测问题中。虽然GMM的乘积仍为GMM，但随着因子数量增加，所得乘积GMM的分量数量呈指数级增长。为了获得单变量因子化概率密度函数（例如GMM乘积）的一个可处理的近似，我们研究了迭代消息传递算法。基于置信传播（Belief Propagation, BP），我们提出了一种基于变量复制与高斯置信传播（Variable Duplication and Gaussian Belief Propagation, VDBP）的算法。VDBP的核心思想是构建一个多元测量模型，其边缘后验等于给定的单变量因子化概率密度函数。随后，我们应用高斯置信传播（Gaussian BP, GaBP）将全局推断问题转化为局部问题。期望传播（Expectation Propagation, EP）是消息传递算法的另一分支。除了将全局近似问题转化为局部问题外，其特点在于包含一个投影操作，确保中间函数（消息）属于期望的分布族。由于这一投影，EP可直接用于近似因子化概率密度函数。然而，即使每个因子均可积，当需要计算一个不可积置信的均值与方差时，EP中的除法操作仍可能导致算法失效。因此，本文提出了两种方法，将EP与我们先前提出的处理不可积置信的技术相结合，以近似单变量因子分布。