Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.

翻译：贝叶斯神经网络为深度学习模型中的推断提供了一种自然的概率论表述。尽管其广受欢迎，但其最优性在统计决策理论的视角下鲜受关注。本文研究了在正态位置模型下、二次损失函数中由深度全连接前馈ReLU贝叶斯神经网络所诱导的决策规则。我们证明，在固定先验尺度下，贝叶斯决策规则并非最小最大。继而，我们提出一种针对贝叶斯神经网络先验有效输出方差超先验，该超先验可产生超调和平方根边缘密度，从而确保所得决策规则同时具备可容许性与最小最大性。我们将这一结果从二次损失场景进一步推广至基于库尔贝克-莱布勒损失的预测密度估计问题。最后，通过数值模拟验证了我们的理论发现。