Simulating Posterior Bayesian Neural Networks with Dependent Weights

arXiv:2507.22095v3 Announce Type: replace
Abstract: We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights, as introduced in Lee et al. 2023, to address limitations of the standard Gaussian prior. It has been proved in Lee et al. 2023 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. Among our results, we present sufficient conditions on the model parameters (the activation function and the associated L’evy measures) which ensure that the sequential limit is independent of the order. Next, 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 L. Carvalho et al. 2025. We illustrate our findings using numerical simulations.