The Metropolis-within-Gibbs (MwG) algorithm is a widely used Markov Chain Monte Carlo method for sampling from high-dimensional distributions when exact conditional sampling is intractable. We study MwG with Random Walk Metropolis (RWM) updates, using proposal variances tuned to match the target's conditional variances. Assuming the target $\pi$ is a $d$-dimensional log-concave distribution with condition number $\kappa$, we establish a spectral gap lower bound of order $\mathcal{O}(1/\kappa d)$ for the random-scan version of MwG, improving on the previously available $\mathcal{O}(1/\kappa^2 d)$ bound. This is obtained by developing sharp estimates of the conductance of one-dimensional RWM kernels, which can be of independent interest. The result shows that MwG can mix substantially faster with variance-adaptive proposals and that its mixing performance is just a constant factor worse than that of the exact Gibbs sampler, thus providing theoretical support to previously observed empirical behavior.

翻译：Metropolis-within-Gibbs（MwG）算法是一种广泛使用的马尔可夫链蒙特卡洛方法，用于从高维分布中采样，此时精确的条件采样难以实现。我们研究了采用随机游走Metropolis（RWM）更新的MwG算法，其建议方差经过调整以匹配目标分布的条件方差。假设目标分布$\pi$是一个条件数为$\kappa$的$d$维log-凹分布，我们为随机扫描版本的MwG建立了阶为$\mathcal{O}(1/\kappa d)$的谱隙下界，改进了先前可用的$\mathcal{O}(1/\kappa^2 d)$界。这一结果是通过对一维RWM核的传导度进行精确估计而获得的，该估计本身可能具有独立的研究价值。结果表明，采用方差自适应建议分布的MwG算法可以显著加快混合速度，且其混合性能仅比精确Gibbs采样器差一个常数因子，从而为先前观察到的经验行为提供了理论支持。