From Shallow Bayesian Neural Networks to Gaussian Processes: General Convergence, Identifiability and Scalable Inference

In this work, we study scaling limits of shallow Bayesian neural networks (BNNs) via their connection to Gaussian processes (GPs), with an emphasis on statistical modeling, identifiability, and scalable inference. We first establish a general convergence result from BNNs to GPs by relaxing assumptions used in prior formulations, and we compare alternative parameterizations of the limiting GP model. Building on this theory, we propose a new covariance function defined as a convex mixture of components induced by four widely used activation functions, and we characterize key properties including positive definiteness and both strict and practical identifiability under different input designs. For computation, we develop a scalable maximum a posterior (MAP) training and prediction procedure using a Nystr"om approximation, and we show how the Nystr"om rank and anchor selection control the cost-accuracy trade-off. Experiments on controlled simulations and real-world tabular datasets demonstrate stable hyperparameter estimates and competitive predictive performance at realistic computational cost.