The posterior over Bayesian neural network (BNN) parameters is extremely high-dimensional and non-convex. For computational reasons, researchers approximate this posterior using inexpensive mini-batch methods such as mean-field variational inference or stochastic-gradient Markov chain Monte Carlo (SGMCMC). To investigate foundational questions in Bayesian deep learning, we instead use full-batch Hamiltonian Monte Carlo (HMC) on modern architectures. We show that (1) BNNs can achieve significant performance gains over standard training and deep ensembles; (2) a single long HMC chain can provide a comparable representation of the posterior to multiple shorter chains; (3) in contrast to recent studies, we find posterior tempering is not needed for near-optimal performance, with little evidence for a "cold posterior" effect, which we show is largely an artifact of data augmentation; (4) BMA performance is robust to the choice of prior scale, and relatively similar for diagonal Gaussian, mixture of Gaussian, and logistic priors; (5) Bayesian neural networks show surprisingly poor generalization under domain shift; (6) while cheaper alternatives such as deep ensembles and SGMCMC methods can provide good generalization, they provide distinct predictive distributions from HMC. Notably, deep ensemble predictive distributions are similarly close to HMC as standard SGLD, and closer than standard variational inference.

研究表明， 使用全批处理的哈密顿蒙特卡罗方法可以提高贝叶斯神经网络的性能，并证明模型中选择的先验分布对性能的影响较小，但相比于深度集成、SGLD 等计算代价较小的方法，HMC 呈现出更接近于精确后验分布的预测分布；同时，研究发现贝叶斯神经网络在领域转移方面表现较差。

贝叶斯神经网络的后验分布是什么样子的？