# Posterior Bayesian Neural Networks with Dependent Weights

arXiv: https://arxiv.org/abs/2507.22095 · HTML: https://mlsanity.com/paper/2507.22095/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 38 references that are in the map.
- First posted on arXiv 29 Jul 2025.

## Builds on (most cited first, 24 of 24)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (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 Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Traditional and Heavy-Tailed Self Regularization in Neural Network Models](https://mlsanity.com/paper/1901.08276/) (2019)
- [Random Neural Networks in the Infinite Width Limit as Gaussian Processes](https://mlsanity.com/paper/2107.01562/) (2021)
- [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)
- [Non-asymptotic approximations of neural networks by Gaussian processes](https://mlsanity.com/paper/2102.08668/) (2021)
- [Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility](https://mlsanity.com/paper/2205.08187/) (2022)
- [Wide Deep Neural Networks with Gaussian Weights are Very Close to Gaussian Processes](https://mlsanity.com/paper/2312.11737/) (2023)
- [The Positivity of the Neural Tangent Kernel](https://mlsanity.com/paper/2404.12928/) (2024)
- [Non-asymptotic approximations of Gaussian neural networks via second-order Poincar'e inequalities](https://mlsanity.com/paper/2304.04010/) (2023)
- [Gaussian random field approximation via Stein's method with applications to wide random neural networks](https://mlsanity.com/paper/2306.16308/) (2023)
- [Deep Stable neural networks: large-width asymptotics and convergence rates](https://mlsanity.com/paper/2108.02316/) (2021)
- [Training Integrable Parameterizations of Deep Neural Networks in the Infinite-Width Limit](https://mlsanity.com/paper/2110.15596/) (2021)
- [Entropic bounds for conditionally Gaussian vectors and applications to neural networks](https://mlsanity.com/paper/2504.08335/) (2025)
- [α-Stable convergence of heavy-tailed infinitely-wide neural networks](https://mlsanity.com/paper/2106.11064/) (2021)
- [Finite-Dimensional Gaussian Approximation for Deep Neural Networks: Universality in Random Weights](https://mlsanity.com/paper/2507.12686/) (2025)
- [Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection](https://mlsanity.com/paper/2407.18707/) (2024)
- [Student-t processes as infinite-width limits of posterior Bayesian neural networks](https://mlsanity.com/paper/2502.04247/) (2025)
- [Function-Space MCMC for Bayesian Wide Neural Networks](https://mlsanity.com/paper/2408.14325/) (2024)

## Abstract (arXiv)

We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in [26], to address limitations of the standard Gaussian prior. It has been proved in [26] that, as the number of nodes in the hidden layers grows large, according to a sequential and ordered limit, the law of the output converges weakly to a Gaussian mixture. In this paper, we study the neural network through the lens of the posterior distribution with a Gaussian likelihood. If the random covariance matrix of the infinite-width limit is positive definite under the prior, we identify the posterior distribution of the output in the wide-width limit according to a sequential regime. Remarkably, we provide mild sufficient conditions to ensure the aforementioned invertibility of the random covariance matrix under the prior, thereby extending the results in [8]. Among our results, we present sufficient conditions on some model parameters (the activation function and the associated L'evy measures) which ensure that the sequential limits are independent of the order. We illustrate our findings with examples and numerical simulations.