Neural Bayesian Networks for Modular and Scalable Decision-Making Under Uncertainty

Human decision-making is sequential and uncertainty-aware, yet standard neural networks often rely on static, dense forward computation with limited visibility into evidence acquisition, uncertainty evolution, or when computation should stop. We introduce Neural Bayesian Sequential Routing (NBSR), a framework that models neural inference as active evidence accumulation over a hierarchical Directed Acyclic Graph (DAG). Within a Dirichlet--Categorical conjugate framework, neural experts query a persistent global knowledge oracle to extract positive evidence vectors, which act as pseudo-counts and update a Dirichlet belief state by exact conjugate addition. Coupled with a Gumbel-Softmax Straight-Through estimator, this update enables hard, path-dependent routing while preserving surrogate gradients for end-to-end training. The resulting Dirichlet precision and entropy provide mechanisms for uncertainty quantification, entropy-based early exiting, OOD abstention, and cost-aware evidence acquisition. We prove that, under strictly positive evidence extraction, total Dirichlet precision increases monotonically along any valid trajectory and marginal predictive variance is bounded, formalizing sequential ``hypothesis sharpening''; under idealized capacity and optimization assumptions, the terminal Dirichlet expectation recovers the Bayes-optimal conditional distribution. Empirical evaluations across visual categorization, structured medical diagnosis, language modeling, partially observable control, and cost-aware Bayesian experimental design show that NBSR achieves competitive predictive performance while providing transparent routing traces, path-dependent evidence attribution, uncertainty-aware decision control, and resource-rational inference. Overall, NBSR offers a mathematically grounded framework for interpretable, modular, and resource-rational agentic AI.

Bayesian neural network predictions are commonly approximated using a fixed number of Monte Carlo samples per input, without controlling the resulting error that comes from this finite sample. We propose the use of confidence sequences to dynamically determine how many samples are needed while maintaining statistical guarantees. We consider several ways in which predictive probabilities are used, including identifying the most likely class, approximating the full predictive distribution, and resolving probability-threshold decisions. Sampling stops once the corresponding decision can be made with the desired guarantee. Experiments show that the method allocates the computational budget efficiently, assigning more samples to ambiguous inputs than to easy inputs while preserving reliable decisions and reducing overall latency relative to a fixed Monte Carlo budget.

Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.