Gaussian Mixture Belief Propagation for Hybrid Inference in Factor Graphs

Sensor data fusion in industrial systems often requires combining continuous measurements with discrete semantic information while accounting for measurement uncertainty and multimodal distributions. We propose Gaussian Mixture Belief Propagation (GMBP), a probabilistic message-passing frame-work for hybrid factor graphs that represents discrete-continuous dependencies using Gaussian mixture models (GMMs). For each discrete state, the corresponding continuous evidence is represented by a conditional GMM, enabling multimodal distributions to be propagated between continuous and discrete variables. In particular, GMBP supports both hard and uncertain continuous evidence and provides closed-form message updates in both directions. The resulting framework enables the fusion of multiple continuous evidence sources into a common discrete hypothesis while retaining the uncertainty of the individual measurements. A synthetic two-class example demonstrates the bidirectional fusion of hard and soft evidence. On a real-world fault classification benchmark, 25 repeated stratified cross-validation runs show that adaptively selecting the Gaussian mixture order per class and feature via Bayesian Information Criterion (BIC) improves accuracy by 5–10 percentage points over both a single-Gaussian baseline and a fixed-order mixture. ... mehrThis demonstrates the benefit of GMBP’s data-driven multimodal representation and highlights the potential of GMM-based belief propagation for probabilistic hybrid sensor data fusion with multimodal continuous evidence.