Convergence of Position-Dependent MALA with Application to Conditional Simulation in GLMMs

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