A Metropolis-Hastings step is widely used for gradient-based Markov chain Monte Carlo methods in uncertainty quantification. By calculating acceptance probabilities on batches, a stochastic Metropolis-Hastings step saves computational costs, but reduces the effective sample size. We show that this obstacle can be avoided by a simple correction term. We study statistical properties of the resulting stationary distribution of the chain if the corrected stochastic Metropolis-Hastings approach is applied to sample from a Gibbs posterior distribution in a nonparametric regression setting. Focusing on deep neural network regression, we prove a PAC-Bayes oracle inequality which yields optimal contraction rates and we analyze the diameter and show high coverage probability of the resulting credible sets. With a numerical example in a high-dimensional parameter space, we illustrate that credible sets and contraction rates of the stochastic Metropolis-Hastings algorithm indeed behave similar to those obtained from the classical Metropolis-adjusted Langevin algorithm.

通过计算关键批次的接受概率，我们展示了在一些应用中通过简单的修正可以避免随机 Metropolis-Hastings 步骤降低有效样本量的障碍。我们在非参数回归背景下运用修正的随机 Metropolis-Hastings 方法，研究了链的稳定分布的统计性质，并通过证明 PAC-Bayes 学习器不等式来获得最优收缩速率，分析了置信集的直径和高覆盖概率。通过高维参数空间中的数值例子，我们展示了随机 Metropolis-Hastings 算法得到的置信集和收缩速率与经典的 Metropolis-adjusted Langevin 算法结果的相似性。

随机Metropolis-Hastings的统计保证