用核函数构造可解释的先验，让神经网络像高斯过程一样量化不确定性。
Bayesian neural networks with interpretable priors from Mercer kernels
- 基于核函数的表示直接定义网络参数先验，实现可解释性。
- 生成的贝叶斯神经网络样本逼近指定高斯过程，保持不确定性建模能力。
- 无需特定网络结构，兼顾高斯过程的可解释性与神经网络的可扩展性。
在科学或工程应用中，数据有限或噪声较大时，准确量化神经网络输出的不确定性至关重要。贝叶斯神经网络（BNN）通过构建网络参数的后验分布来实现这一目标，但其先验通常缺乏实际意义。这是因为神经网络的输入-输出映射过于复杂，难以理解特定分布如何对输出空间施加可解释的约束。相比之下，高斯过程（GPs）因具备可解释性常被用于不确定性量化，但其在大规模数据集上受限，且依赖于协方差核具有特定结构。为此，本文提出一种新型的BNN先验——梅尔科夫先验（Mercer priors），使生成的BNN样本近似于指定的高斯过程。该方法通过梅尔科夫表示直接在参数空间定义先验，不依赖网络的具体结构，从而在保持可扩展性的基础上实现有意义的贝叶斯建模。
原文摘要 · Abstract (English)
Quantifying the uncertainty in the output of a neural network is essential for deployment in scientific or engineering applications where decisions must be made under limited or noisy data. Bayesian neural networks (BNNs) provide a framework for this purpose by constructing a Bayesian posterior distribution over the network parameters. However, the prior, which is of key importance in any Bayesian setting, is rarely meaningful for BNNs. This is because the complexity of the input-to-output map of a BNN makes it difficult to understand how certain distributions enforce any interpretable constraint on the output space of the network. Gaussian processes (GPs), on the other hand, are often preferred in uncertainty quantification tasks due to their interpretability. The drawback is that GPs are limited to small datasets without advanced techniques, which often rely on the covariance kernel having a specific structure. To address these challenges, we introduce a new class of priors for BNNs, called Mercer priors, such that the resulting BNN has samples which approximate that of a specified GP. The method works by defining a prior directly over the network parameters from the Mercer representation of the covariance kernel, and does not rely on the network having a specific structure. In doing so, we can exploit the scalability of BNNs in a meaningful Bayesian way.
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