Neural Bayesian Sequential Routing

We introduce Neural Bayesian Sequential Routing (NBSR), a framework that turns neural inference into a sequential Bayesian decision process. NBSR combines four key ideas: a persistent global knowledge oracle, a hierarchical graph of routers and experts, an explicit Dirichlet belief state updated by accumulated neural evidence, and hard belief-conditioned routing trained end-to-end with Gumbel-Softmax. Together, these components unify conditional computation, uncertainty estimation, and sequential decision-making in one architecture. The evolving belief state enables early exit, OOD abstention, cost-aware evidence acquisition, and transparent path-wise audit trails. We prove monotone growth of Dirichlet precision and bounded predictive variance under positive evidence, formalizing sequential hypothesis sharpening, and show Bayes-optimal recovery under idealized assumptions. Extensions with recurrent memory and autoregressive routing support partially observable control and Bayesian experimental design. Across vision, diagnosis, language modeling, control, and active triage, NBSR achieves competitive predictive performance while providing uncertainty-aware, interpretable, and resource-rational inference.