One main limitation of the existing optimal scaling results for Metropolis--Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate $0.234$ can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.

中文翻译：

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随机游走都市算法在一般目标分布上的最佳缩放

Metropolis--Hastings 算法现有最优缩放结果的一个主要限制是对目标分布的假设是不切实际的。在本文中，我们考虑了随机游走 Metropolis 算法在高维的一般目标分布上的最佳缩放，这些分布来自贝叶斯统计的实际 MCMC 模型。为了通过最大化预期平方跳跃距离 (ESJD) 进行优化缩放，我们表明在目标分布的一般现实充分条件下可以获得渐近最优接受率 $0.234$。新的充分条件很容易被验证，并且可能适用于由贝叶斯统计应用产生的一些一般类别的 MCMC 模型，这些模型基本上概括了大多数现有最佳缩放文献中所需的乘积 iid 条件。此外，我们表明可以在稍强的条件下获得一维扩散限制，这仍然允许目标分布的相关坐标。我们还将新的扩散限制结果与高维 Metropolis 算法的复杂性边界联系起来。