Approximating univariate factored distributions via message-passing algorithms

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. arXiv:2602.01377v2  [eess.SP]  3 Feb 2026 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.