Directional Metropolis-Hastings

TL;DR

The paper introduces Directional Metropolis-Hastings, a new MCMC kernel with state-dependent covariance that uses target distribution derivatives to improve proposal plausibility, with proven ergodicity and adaptive updates.

Contribution

It presents a novel proposal kernel for Metropolis-Hastings that adapts orientation based on target derivatives, along with ergodicity conditions and adaptive variance schemes.

Findings

Enhanced proposal plausibility through directional adaptation.

Proven geometric ergodicity under specified conditions.

Improved performance in Bayesian generalized linear models.

Abstract

We propose a new kernel for Metropolis Hastings called Directional Metropolis Hastings (DMH) with multivariate update where the proposal kernel has state dependent covariance matrix. We use the derivative of the target distribution at the current state to change the orientation of the proposal distribution, therefore producing a more plausible proposal. We study the conditions for geometric ergodicity of our algorithm and provide necessary and sufficient conditions for convergence. We also suggest a scheme for adaptively update the variance parameter and study the conditions of ergodicity of the adaptive algorithm. We demonstrate the performance of our algorithm in a Bayesian generalized linear model problem.