Laplace approximations are popular techniques for endowing deep networks with epistemic uncertainty estimates as they can be applied without altering the predictions of the neural network, and they scale to large models and datasets. While the choice of prior strongly affects the resulting posterior distribution, computational tractability and lack of interpretability of weight space typically limit the Laplace approximation to isotropic Gaussian priors, which are known to cause pathological behavior as depth increases. As a remedy, we directly place a prior on function space. More precisely, since Lebesgue densities do not exist on infinite-dimensional function spaces, we have to recast training as finding the so-called weak mode of the posterior measure under a Gaussian process (GP) prior restricted to the space of functions representable by the neural network. Through the GP prior, one can express structured and interpretable inductive biases, such as regularity or periodicity, directly in function space, while still exploiting the implicit inductive biases that allow deep networks to generalize. After model linearization, the training objective induces a negative log-posterior density to which we apply a Laplace approximation, leveraging highly scalable methods from matrix-free linear algebra. Our method provides improved results where prior knowledge is abundant, e.g., in many scientific inference tasks. At the same time, it stays competitive for black-box regression and classification tasks where neural networks typically excel.

我们提出一种基于拉普拉斯近似和高斯过程先验的方法，通过直接在函数空间中施加先验来解决深度网络中整体性误差估计的问题，并通过矩阵自由线性代数的高度可扩展方法获得改进的结果。

FSP-Laplace：贝叶斯深度学习中拉普拉斯近似的函数空间先验