We develop necessary conditions for geometrically fast convergence in the Wasserstein distance for Metropolis-Hastings algorithms on $\mathbb{R}^d$ when the metric used is a norm. This is accomplished through a lower bound which is of independent interest. We show exact convergence expressions in more general Wasserstein distances (e.g. total variation) can be achieved for a large class of distributions by centering an independent Gaussian proposal, that is, matching the optimal points of the proposal and target densities. This approach has applications for sampling posteriors of many popular Bayesian generalized linear models. In the case of Bayesian binary response regression, we show when the sample size $n$ and the dimension $d$ grow in such a way that the ratio $d/n \to \gamma \in (0, +\infty)$, the exact convergence rate can be upper bounded asymptotically.

翻译：我们为大都会-哈斯廷算法的瓦塞斯坦距离的几何性快速趋同发展了必要的条件,如果所使用的衡量标准是一种规范,则该算法以$\mathbb{R ⁇ d$为标准。这是通过一个独立感兴趣的较低界限实现的。我们在更普遍的瓦塞斯坦距离(例如,总变异)中显示精确的趋同表达方式,可以通过将独立的高萨建议集中到大宗分配类别,即匹配提案的最佳点和目标密度。这个方法可应用许多流行的巴伊西亚通用线性模型的后方取样。在巴伊西亚二元反应回归中,当样本大小为$n和维度为$d$增长时,我们显示,美元/n 至\ gamma\ in (0, ⁇ fty) 的比,准确的趋同率可以被上下以中断线方式约束。