We study general coordinate-wise MCMC schemes (such as Metropolis-within-Gibbs samplers), which are commonly used to fit Bayesian non-conjugate hierarchical models. We relate their convergence properties to the ones of the corresponding (potentially not implementable) Gibbs sampler through the notion of conditional conductance. This allows us to study the performances of popular Metropolis-within-Gibbs schemes for non-conjugate hierarchical models, in high-dimensional regimes where both number of datapoints and parameters increase. Given random data-generating assumptions, we establish dimension-free convergence results, which are in close accordance with numerical evidences. Applications to Bayesian models for binary regression with unknown hyperparameters and discretely observed diffusions are also discussed. Motivated by such statistical applications, auxiliary results of independent interest on approximate conductances and perturbation of Markov operators are provided.

翻译：本文研究通用的坐标式MCMC方案（如Metropolis-within-Gibbs采样器），这类方案常用于拟合贝叶斯非共轭层次模型。我们通过条件电导的概念，将其收敛性质与对应（可能无法实际实现的）Gibbs采样器的收敛特性联系起来。这使我们能够研究在数据点数量和参数数量均增长的高维情形下，针对非共轭层次模型的常用Metropolis-within-Gibbs方案的性能。在随机数据生成假设下，我们建立了与维度无关的收敛结果，该结果与数值证据高度吻合。本文还讨论了该方法在具有未知超参数的二元回归贝叶斯模型及离散观测扩散过程中的应用。受此类统计应用的启发，我们还提供了关于近似电导与马尔可夫算子扰动的独立辅助结果。