Statistical guarantees for stochastic Metropolis-Hastings

• URL: http://arxiv.org/abs/2310.09335v1
• Date: Fri, 13 Oct 2023 18:00:26 GMT
• Title: Statistical guarantees for stochastic Metropolis-Hastings
• Authors: Sebastian Bieringer, Gregor Kasieczka, Maximilian F. Steffen and Mathias Trabs
• Abstract summary: By calculating acceptance probabilities on batches, a 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 show that the Metropolis-Hastings algorithm indeed behave similar to those obtained from the classical Metropolis-adjusted Langevin algorithm.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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