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From the classical and influential works of Neal (1996), it is known that theinfinite width scaling limit of a Bayesian neural network with one hidden layeris a Gaussian process, \emph{when the network weights have bounded priorvariance}. Neal's result has been extended to networks with multiple hiddenlayers and to convolutional neural networks, also with Gaussian process scalinglimits. The tractable properties of Gaussian processes then allowstraightforward posterior inference and uncertainty quantification,considerably simplifying the study of the limit process compared to a networkof finite width. Neural network weights with unbounded variance, however, poseunique challenges. In this case, the classical central limit theorem breaksdown and it is well known that the scaling limit is an $\alpha$-stable processunder suitable conditions. However, current literature is primarily limited toforward simulations under these processes and the problem of posteriorinference under such a scaling limit remains largely unaddressed, unlike in theGaussian process case. To this end, our contribution is an interpretable andcomputationally efficient procedure for posterior inference, using a\emph{conditionally Gaussian} representation, that then allows full use of theGaussian process machinery for tractable posterior inference and uncertaintyquantification in the non-Gaussian regime.

Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks under Weights with Unbounded Variance