SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1765-A1788, January 2020.
Markov chain Monte Carlo (MCMC) samplers are numerical methods for drawing samples from a given target probability distribution. We discuss one particular MCMC sampler, the MALA-within-Gibbs sampler, from the theoretical and practical perspectives. We first show that the acceptance ratio and step size of this sampler are independent of the overall problem dimension when (i) the target distribution has sparse conditional structure, and (ii) this structure is reflected in the partial updating strategy of MALA-within-Gibbs. If, in addition, the target density is blockwise log-concave, then the sampler's convergence rate is independent of dimension. From a practical perspective, we expect that MALA-within-Gibbs is useful for solving high-dimensional Bayesian inference problems where the posterior exhibits sparse conditional structure at least approximately. In this context, a partitioning of the state that correctly reflects the sparse conditional structure must be found, and we illustrate this process in two numerical examples. We also discuss trade-offs between the block size used for partial updating and computational requirements that may increase with the number of blocks.

中文翻译：

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具有稀疏条件结构的高维分布的MALA-in-gibbs采样器

SIAM科学计算杂志，第42卷，第3期，第A1765-A1788页，2020年1月。
马尔可夫链蒙特卡洛（MCMC）采样器是从给定目标概率分布中抽取样本的数值方法。我们将从理论和实践的角度讨论一种特定的MCMC采样器，即MALA-in-Gibbs采样器。我们首先证明，当（i）目标分布具有稀疏的条件结构，并且（ii）该结构反映在MALA的部分更新策略中时，该采样器的接受率和步长与整体问题维度无关。吉布斯。此外，如果目标密度是逐块对数凹的，则采样器的收敛速度与尺寸无关。从实际的角度来看，我们希望，MALA-in-Gibbs可用于解决高维贝叶斯推断问题，其中后验至少显示出稀疏的条件结构。在这种情况下，必须找到正确反映稀疏条件结构的状态划分，并且我们在两个数值示例中说明了此过程。我们还将讨论用于部分更新的块大小和可能随块数而增加的计算需求之间的权衡。