Gaussian Mixture Belief Propagation for Probabilistic Graphical Models

Inference in continuous probabilistic graphical models remains challenging due to nonlinear dependencies, multi-modal posteriors, and the absence of closed-form solutions. While Gaussian Belief Propagation (GaBP) offers efficient inference for unimodal Gaussians, it fails to capture complex, data-driven distributions. Sampling based methods, in turn, lack analytical tractability and scalability. We present Gaussian Mixture Belief Propagation (GMBP), a novel framework that represents pair-wise dependencies as joint Gaussian mixture models (GMMs) and enables closed-form message passing via analytic conditioning and marginalization. GMBP unifies the treatment of hard (Dirac) and soft (Gaussian) evidence, updating each mixture component in a Kalman-style fashion while preserving multimodality. To prevent the combinatorial explosion of components, we integrate principled Gaussian mixture reduction (GMR) techniques, ensuring computational tractability without sacrificing expressive power. Evaluated on synthetic graphs, the Abalone benchmark dataset, and real-world material science data from the Circular Factory, GMBP demonstrates superior likelihood and multimodal fidelity compared to GaBP. ... mehrOur results establish GMBP as a flexible, scalable, and analytically grounded inference engine for continuous factor graphs in data-rich, nonlinear domains.