Using the framework of weak Poincaré inequalities, we analyze the convergence properties of deterministic-scan Metropolis-within-Gibbs samplers, an important class of Markov chain Monte Carlo algorithms. Our analysis applies to nonreversible Markov chains and yields explicit (subgeometric) convergence bounds through novel comparison techniques based on Dirichlet forms. We show that the joint chain inherits the convergence behavior of the marginal chain and conversely. In addition, we establish several fundamental results for weak Poincaré inequalities for discrete-time Markov chains, such as a tensorization property for independent chains. We apply our theoretical results through applications to algorithms for Bayesian inference for a hierarchical regression model and a diffusion model under discretely-observed data.

翻译：利用弱Poincaré不等式框架，我们分析了一类重要的马尔可夫链蒙特卡洛算法——确定性扫描Metropolis-within-Gibbs采样器的收敛性质。我们的分析适用于不可逆马尔可夫链，并通过基于狄利克雷形式的新颖比较技术，得到了显式的（次几何）收敛界。我们证明了联合链继承了边缘链的收敛行为，反之亦然。此外，我们建立了离散时间马尔可夫链弱Poincaré不等式的若干基本结果，例如独立链的张量化性质。我们通过将理论结果应用于分层回归模型和离散观测数据下扩散模型的贝叶斯推断算法，展示了其实际应用价值。