Direct Message Approximation (DMA): A Consistency-Based Framework for Tractable Approximate Inference on Factor Graphs

Approximate message passing on factor graphs underlies two dominant families of probabilistic inference algorithms: expectation propagation (EP) and variational message passing (VMP). Both methods approximate the marginal at each factor edge, forcing an iterative round-robin schedule, risking negative-precision messages, and, for VMP, collapsing to point estimates at Dirac-delta factors. We introduce Direct Message Approximation (DMA), which approximates factor-to-variable messages directly rather than the marginal. For normalisable factors, we define a consistency condition (requiring exactness when all other incoming messages are Dirac deltas) to guide message construction. We prove a master theorem (proper messages, any graph) bounding marginal KL from message KL, with three structural corollaries: Dirac-input consistency, no EP-style inner-loop iteration, and no negative-precision messages. Further, we prove a complementary O(1/r2) guarantee for the inherently improper backward message of the product factor, whose closed-form treatment has resisted prior work. As a concrete instantiation, we derive explicit DMA messages for the product and leaky-ReLU factors and assemble a Bayesian neural network (BNN) inference algorithm with one forward/backward sweep per training example and no gradient learning-rate hyperparameter, validating that the structural guarantees translate to predictive uncertainty that widens in data-sparse regions, including under model mismatch.