We establish verifiable 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 variants of MALA like the manifold MALA (MMALA), the proposal covariance matrix changes in every iteration. Thus, we provide conditions for geometric ergodicity of different variations of the 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 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.
翻译:我们建立了可核查的条件,使大都会-哈斯廷斯(MH)算法(MH)具有依赖位置的建议共变矩阵,这种算法具有或不会具有几何趋同率。一些基于扩散的MH算法,如大都会调整的朗埃文算法(MALA)和预先设定的MAMALA(PCMALA),具有一个独立位置的建议差异。而对于MALA(MMALA)等多种变体的其他变体,如MALA(MMAALA),建议共变式矩阵在每次迭代中的变化。因此,我们为朗埃文算法不同变异的几何异性提供了条件。在两种最受欢迎的普遍线性混合模型(GLMM)的有条件模拟(GLMM)中验证了这些条件,即与日志链接的二元GLMMMM和与日志链接的Poisson GLMMM。一些空间GLMMM(M)框架中的“经验性比较”表明,计算成本较低的PCMALA与适当选择的预调控矩阵可能优于MALA的MALA。