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 Lévy 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.

翻译：我们考虑具有依赖性和可能重尾权重的全连接前馈深度神经网络，如Lee等人（2023）所引入，以解决标准高斯先验的局限性。Lee等人（2023）已证明，当隐藏层节点数量趋于无穷时，根据顺序有序极限，输出的分布弱收敛于高斯混合分布。在我们的结果中，我们提出了模型参数（激活函数及相关Lévy测度）的充分条件，以确保顺序极限与次序无关。接着，我们通过高斯似然下的后验分布来研究神经网络。若无限宽度极限的随机协方差矩阵在先验下正定，我们根据顺序极限识别了宽宽度极限下输出的后验分布。值得注意的是，我们提供了温和的充分条件以确保先验下该随机协方差矩阵的可逆性，从而扩展了L. Carvalho等人（2025）的结果。我们通过数值模拟展示了我们的发现。