# Student-t processes as infinite-width limits of posterior Bayesian neural networks

arXiv: https://arxiv.org/abs/2502.04247 · HTML: https://mlsanity.com/paper/2502.04247/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 23 references that are in the map.
- First posted on arXiv 6 Feb 2025.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Student-t Processes as Alternatives to Gaussian Processes](https://mlsanity.com/paper/1402.4306/) (2014)
- [Exact posterior distributions of wide Bayesian neural networks](https://mlsanity.com/paper/2006.10541/) (2020)
- [Quantitative Gaussian Approximation of Randomly Initialized Deep Neural Networks](https://mlsanity.com/paper/2203.07379/) (2022)
- [Quantitative CLTs in Deep Neural Networks](https://mlsanity.com/paper/2307.06092/) (2023)
- [Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processes](https://mlsanity.com/paper/2312.11737/) (2023)
- [Upgrading from Gaussian Processes to Student's-T Processes](https://mlsanity.com/paper/1801.06147/) (2018)

## Built on by (most cited first, 3 of 3)

- [Uncertainty-Weighted Fusion of Image and Synthetic Event for Video Anomaly Detection](https://mlsanity.com/paper/2505.02393/) (2025)
- [Posterior Bayesian Neural Networks with Dependent Weights](https://mlsanity.com/paper/2507.22095/) (2025)
- [From Overfitting to Reliability: Introducing the Hierarchical Approximate Bayesian Neural Network](https://mlsanity.com/paper/2512.13111/) (2025)

## Abstract (arXiv)

The asymptotic properties of Bayesian Neural Networks (BNNs) have been extensively studied, particularly regarding their approximations by Gaussian processes in the infinite-width limit. We extend these results by showing that posterior BNNs can be approximated by Student-t processes, which offer greater flexibility in modeling uncertainty. Specifically, we show that, if the parameters of a BNN follow a Gaussian prior distribution, and the variance of both the last hidden layer and the Gaussian likelihood function follows an Inverse-Gamma prior distribution, then the resulting posterior BNN converges to a Student-t process in the infinite-width limit. Our proof leverages the Wasserstein metric to establish control over the convergence rate of the Student-t process approximation.