Direct Bethe Free Energy Minimization for Bayesian Neural Networks

Bayesian neural networks are typically trained on the evidence lower bound (ELBO), which keeps the joint likelihood but pays a Jensen gap at every observation. We train by local consistency instead: direct minimisation of the Bethe free energy, whose data term pays no gap-it scores each observation exactly by its predictive density, a strictly proper rule, for any likelihood with a tractable predictive convolution. Our departure is free routing: the beliefs are trained as free parameters of this objective, jointly with the backbone, rather than bound to the conjugate posterior computed in closed form (closed routing). Instantiated with a Gaussian last layer over a deterministic backbone, exact inference appears as the known, closed-routed corner-the neural-linear marginal likelihood, evidence-optimal, keeping the joint. A shared cavity instead trades it for a batchable per-plate predictive score, and free routing reaches that score's optimum-unattainable under the binding whenever the noise is heteroscedastic-improving NLL and calibration over the exact corner. This instance, SCROLL (Shared-Cavity fRee-rOuting Last-Layer), is a single-pass, any-likelihood Bayesian neural network, implicitly empirical-Bayes-prior precision, observation noise, covariance, and backbone fit in one gradient pass. At a single training run and forward pass per architecture-where the conventional references cross-validate their regularisation weight and ensembles pay 5-50x at inference-a fixed SCROLL variant is best-or-tied on NLL and calibration on 7/8 UCI regression benchmarks, and best on 4/5 across three large tabular datasets and two frozen text/vision embeddings.