arXiv:2610.13146v1 Announce Type: new Abstract: Belief propagation (BP) is widely used for inference on probabilistic graphical models but often suffers from non-convergence and poor accuracy on loopy factor graphs, yielding inexact variance estimates in the Gaussian case. Generalized BP (GBP) improves accuracy but relies on manual design or application-specific heuristics for region-graph construction. We propose extended neighborhood message passing (ENMP), which automatically constructs extended neighborhoods of variable nodes from the bipartite factor graph topology, capturing loop-induced correlations without manual region design. The framework applies to general distributions on factor graphs with a tunable accuracy-complexity trade-off. In the exact regime, message updates converge to the unique fixed point of the ENMP equations in finitely many iterations, yielding exact marginals; in the approximate regime, ENMP provides approximate marginals. For Gaussian factor graphs, we derive closed-form update rules enabling exact variance estimation in the exact regime, addressing the long-standing inaccurate variance estimates of loopy Gaussian BP. We evaluate ENMP on densely loopy Gaussian factor graphs from power-grid benchmarks and on a discrete Ising factor graph. Results show accuracy improves with neighborhood size, with exact marginals achieved on several systems in both settings. On larger Gaussian systems, ENMP outperforms regularized Gaussian BP (sGaBP-m) in both mean and variance estimation accuracy.