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

arXiv: https://arxiv.org/abs/2108.12662 · HTML: https://mlsanity.com/paper/2108.12662/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2021.

## Built on by (most cited first, 5 of 5)

- [Harnessing the Power of Reinforcement Learning for Adaptive MCMC](https://mlsanity.com/paper/2507.00671/) (2025)
- [A Targeted Accuracy Diagnostic for Variational Approximations](https://mlsanity.com/paper/2302.12419/) (2023)
- [Gaussian Invariant Markov Chain Monte Carlo](https://mlsanity.com/paper/2506.21511/) (2025)
- [Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlo](https://mlsanity.com/paper/2301.01409/) (2023)
- [Stop the Sampler! Classifier-Based Adaptive Stopping for Sampling Kernels](https://mlsanity.com/paper/2606.16073/) (2026)

## Abstract (arXiv)

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.