Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks under Weights with Unbounded Variance - read online free

Aug 2, 2026

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From Neal’s (1996) result, the infinite-width limit of a Bayesian neural network with one hidden layer becomes a Gaussian process when weights have bounded prior variance. Extensions cover multiple hidden layers and convolutional architectures, enabling tractable posterior inference and uncertainty quantification via Gaussian process machinery. Unbounded weight variance breaks the classical central limit theorem, yielding an α-stable scaling limit. The work addresses posterior inference in this non-Gaussian regime using a conditionally Gaussian representation for interpretable, computationally feasible inference.

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This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Loría, Jorge; Bhadra, Anindya Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks under Weights with Unbounded Variance Published in: Proceedings of Machine Learning Research Published: 01/01/2024 Document Version Publisher's PDF, also known as Version of record Published under the following license: CC BY Please cite the original version: Loría, J. , & Bhadra, A. (2024) . Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks under Weights with Unbounded Variance. Proceedings of Machine Learning Research, 244 , 2331-2349. [https://proceedings.mlr.press/v244/loria24a.html](https://proceedings.mlr.press/v244/loria24a.html) This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you foryour research use or educational purposes in electronic or print form. You must obtain permission for anyother use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Posterior Inference on Shallow Infinitely Wide Bayesian Neural Networks under Weights with Unbounded Variance Jorge Loría 1,2 Anindya Bhadra 1 1Department of Statistics, Purdue University, West Lafayette, Indiana, USA 2Department of Computer Science, Aalto University, Finland Abstract From the classical and influential works of Neal (1996), it is known that the infinite width scaling limit of a Bayesian neural network with one hidden layer is a Gaussian process, when the network weights have bounded prior variance. Neal’s result has been extended to networks with multiple hidden layers and to convolutional neural networks, also with Gaussian process scaling limits. The tractable properties of Gaussian processes then allow straightforward posterior inference and uncertainty quantification, considerably simplifying the study of the limit process compared to a network of finite width. Neural network weights with unbounded variance, however, pose unique challenges. In this case, the classical central limit theorem breaks down and it is well known that the scaling limit is an α-stable process under suitable conditions. However, current literature is primarily limited to forward simulations under these processes and the problem of posterior inference under such a scaling limit remains largely unaddressed, unlike in the Gaussian process case. To this end, our contribution is an interpretable and computationally feasible procedure for posterior inference, using a conditionally Gaussian representation, that then allows full use of the Gaussian process machinery for tractable posterior inference and uncertainty quantification in the non-Gaussian regime. 1 INTRODUCTION Gaussian processes (GPs) have been studied as the infinite width limit of Bayesian neural networks with priorson network weights that have finite variance (Neal, 1996 ; Williams, 1996) . This presents some key advantages over Bayesian neural networks with finite widths that usually require computation intensive Markov chain Monte Carlo (MCMC) posterior calculations (Neal, 1996) or variational approximations (Goodfellow et al., 2016, Chapter 19); in contrast to straightforward posterior inference and probabilistic uncertainty quantification afforded by the GP machinery (Williams and Rasmussen, 2006) . In this sense, the work of Neal (1996) is foundational. The technical reason for this convergence to a GP is due to an application of the central limit theorem under the bounded second moment condition. More specifically, given an I dimensional input x and a one-dimensional output y(x), a K layer feedforward deep neural network (DNN) with K − 1 hidden layers is defined by the recursion: zj(l+1)(

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How does the paper enable posterior inference under an α-stable scaling limit?

It proposes an interpretable and computationally feasible procedure for posterior inference using a conditionally Gaussian representation. This representation allows leveraging Gaussian process tools for tractable posterior inference and uncertainty quantification in the non-Gaussian regime.

Aug 2, 2026

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