Convergence of position-dependent MALA with application to conditional simulation in GLMMs

TL;DR

This paper analyzes the convergence properties of position-dependent MALA algorithms, providing conditions for geometric ergodicity and demonstrating their practical implications in generalized linear mixed models, with empirical comparisons showing PCMALA's efficiency.

Contribution

It establishes conditions for the geometric ergodicity of various MALA algorithms, including position-dependent variants like MMALA and PCMALA, in the context of GLMMs.

Findings

PCMALA can outperform MMALA in spatial GLMMs.

Conditions for geometric ergodicity depend on the proposal covariance structure.

Empirical results support the practical use of PCMALA with appropriate pre-conditioning.

Abstract

We establish conditions under which Metropolis-Hastings (MH) algorithms with a position-dependent proposal covariance matrix will or will not have the geometric rate of convergence. Some of the diffusions based MH algorithms like the Metropolis adjusted Langevin algorithm (MALA) and the pre-conditioned MALA (PCMALA) have a position-independent proposal variance. Whereas, for other modern variants of MALA like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for geometric ergodicity of different variations of the Langevin algorithms. These results have important practical implications as these provide crucial justification for the use of asymptotically valid Monte Carlo standard errors for Markov chain based estimates. The general conditions are verified in the context of conditional simulation from the two most popular generalized linear mixed models (GLMMs), namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. Empirical comparison in the framework of some spatial GLMMs shows that the computationally less expensive PCMALA with an appropriately chosen pre-conditioning matrix may outperform the MMALA.