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 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.本文确立了带位置依赖提议协方差矩阵的梅特罗波利斯-黑斯廷斯（Metropolis-Hastings, MH）算法具备或不具备几何收敛速率的条件。部分基于扩散的MH算法，如梅特罗波利斯调整朗之万算法（Metropolis adjusted Langevin algorithm, MALA）与预条件MALA（pre-conditioned MALA, PCMALA），其提议方差与位置无关；而诸如适配目标分布几何特性的流形MALA（manifold MALA, MMALA）这类现代MALA变体，其提议协方差矩阵在每次迭代中均会发生变化。据此，本文给出了各类朗之万算法变体的几何遍历性条件。上述结论具备重要的实际意义，可为基于马尔可夫链的估计中使用渐近有效的蒙特卡洛标准误提供关键依据。本文在两类最主流的广义线性混合模型（generalized linear mixed models, GLMMs）——即带logit连接函数的二项式GLMM与带log连接函数的泊松GLMM——的条件模拟场景中验证了一般性条件的有效性。针对部分空间GLMM的实证对比表明，选用恰当预条件矩阵的计算成本更低的PCMALA，其性能或优于MMALA。