Kernel Adaptive Metropolis-Hastings

TL;DR

The paper introduces a kernel-based adaptive Metropolis-Hastings algorithm that efficiently samples from complex, nonlinear distributions without requiring gradient information, outperforming existing methods in various challenging scenarios.

Contribution

It presents a novel kernel embedding approach for adaptive MCMC that adapts proposals based on local covariance without gradients, suitable for complex target distributions.

Findings

Outperforms fixed and adaptive samplers on nonlinear targets

Does not require gradient or higher-order information

Efficient and easy to implement in practice

Abstract

A Kernel Adaptive Metropolis-Hastings algorithm is introduced, for the purpose of sampling from a target distribution with strongly nonlinear support. The algorithm embeds the trajectory of the Markov chain into a reproducing kernel Hilbert space (RKHS), such that the feature space covariance of the samples informs the choice of proposal. The procedure is computationally efficient and straightforward to implement, since the RKHS moves can be integrated out analytically: our proposal distribution in the original space is a normal distribution whose mean and covariance depend on where the current sample lies in the support of the target distribution, and adapts to its local covariance structure. Furthermore, the procedure requires neither gradients nor any other higher order information about the target, making it particularly attractive for contexts such as Pseudo-Marginal MCMC. Kernel Adaptive Metropolis-Hastings outperforms competing fixed and adaptive samplers on multivariate, highly nonlinear target distributions, arising in both real-world and synthetic examples. Code may be downloaded at https://github.com/karlnapf/kameleon-mcmc.