A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

中文翻译：

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连续马尔可夫链亚稳态的简单条件

一个家庭$\{Q_{\beta}\}_{\beta \geq 0}$马尔可夫链据说展示亚稳态混合和模式$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$如果它的光谱间隙（或其他一些混合特性）非常接近最差电导$\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^ {(k)}\大)\大)$它的所有大值的模式$\beta$. 我们给出了马尔可夫链族在这个意义上表现出亚稳态的简单充分条件，并验证了这些条件是否适用于针对混合分布的原型 Metropolis-Hastings 链。现有的亚稳态文献很多，我们目前的工作旨在填补以下小空白：使用经过充分研究的方法从统计学中找到易于验证的典型亚稳态的充分条件，同时给出渐近精确的光谱间隙的公式（而不是可能非常远离尖锐的界限）。我们从这篇论文中得到的界限被用在了一篇配套论文中（O. Mangoubi、NS Pillai 和 A. Smith，arXiv:1808.03230) 比较哈密顿蒙特卡罗算法和随机游走算法的多模态目标分布的混合时间。