混合与合成下的均匀遍历性与几何遍历性--广义马尔可夫链蒙特卡罗
Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition
摘要：
在马尔可夫链蒙特卡罗中，通常的做法是每次更新一个变量（或变量的子块），而不是进行单个全维更新。当可以从与目标相关联的每个全条件分布中提取时，这仅仅是吉布斯采样器。通常至少有一个吉布斯更新被替换为Metropolis-Hastings步骤，产生Metropolis-Hastings-within-Gibbs算法。组合组件更新的策略包括组合、随机序列和随机扫描。虽然这些策略可以简化MCMC的实现，并产生优越的经验性能相比，全维更新，相关的马尔可夫链的理论收敛性能受到了有限的关注。我们提出的条件下，一些组件明智的马尔可夫链收敛到平稳分布的几何速度。我们特别注意的各种组件明智的策略的收敛速度之间的连接。这一点很重要，因为它确保了MCMC从业者可以使用的工具的存在，以便对仿真结果充满信心，就好像它们是基于独立和相同分布的样本一样。我们说明我们的结果在两个例子，包括分层线性混合模型和一个涉及极大似然估计的混合模型。
英文摘要：
It is common practice in Markov chain Monte Carlo to update the simulation one variable (or sub-block of variables) at a time, rather than conduct a single full-dimensional update. When it is possible to draw from each full-conditional distribution associated with the target this is just a Gibbs sampler. Often at least one of the Gibbs updates is replaced with a Metropolis-Hastings step, yielding a Metropolis-Hastings-within-Gibbs algorithm. Strategies for combining component-wise updates include composition, random sequence and random scans. While these strategies can ease MCMC implementation and produce superior empirical performance compared to full-dimensional updates, the theoretical convergence properties of the associated Markov chains have received limited attention. We present conditions under which some component-wise Markov chains converge to the stationary distribution at a geometric rate. We pay particular attention to the connections between the convergence rates of the various component-wise strategies. This is important since it ensures the existence of tools that an MCMC practitioner can use to be as confident in the simulation results as if they were based on independent and identically distributed samples. We illustrate our results in two examples including a hierarchical linear mixed model and one involving maximum likelihood estimation for mixed models.
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