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

We establish verifiable conditions under which Metropolis Hastings (MH) algorithms with position-dependent proposal covariance matrix will or will not have geometric rate of convergence. Some of the diffusions based MH algorithms like Metropolis adjusted Langevin algorithms (MALA) and Pre-conditioned MALA (PCMALA) have position independent proposal variance. Whereas, for other variants of MALA like manifold MALA (MMALA), the proposal covariance matrix changes in every iteration. Thus, we provide conditions for geometric ergodicity of different variations of Langevin algorithms. These conditions are verified in the context of conditional simulation from the two most popular generalized linear mixed models (GLMMs), namely the binomial GLMM with logit link and the Poisson GLMM with log link. Empirical comparison in the framework of some spatial GLMMs shows that computationally less expensive PCMALA with an appropriately chosen pre-conditioning matrix may outperform MMALA.