A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants
A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants
复制标题
跟踪类数据增强算法及其三明治变体的谱分析比较
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
J. Hobert
中科院分区：
文献类型：
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作者：
K. Khare;J. Hobert
数据增强（DA）算法是一种广泛使用的马尔可夫链蒙特卡罗算法，易于实现，但往往收敛缓慢。三明治算法是一种替代方案，它可以更快地收敛，同时每次迭代需要大致相同的计算工作量。从理论上讲，三明治算法总是至少与相应的DA算法一样快地收敛，即$\Vert {K^*}\Vert \le \Vert {K}\Vert$，其中$K$和$K^*$分别是与DA和三明治算法相关的Markov运算符，$\Vert\cdot\Vert$表示运算符范数。本文给出了这个算子范数不等式的一个实质性的改进。特别地，在正则性条件下暗示$K$是一个迹类算子，它被证明$K^*$也是一个正的迹类算子，并且$K^*$的谱优于$K$的谱，因为前者的有序元素都小于或等于后者的相应元素。此外，如果三明治算法是使用群作用构造的，如Liu和Wu [J. Amer. Statist. Assoc.94（1999）1264- 1274]和Hobert和Marchev [Ann. Statist. 36（2008）532- 554]，则在至少一对特征值之间存在严格不等式。这些结果被应用到一个新的DA算法的贝叶斯分位数回归介绍Kozumi和小林[J.统计。Comput.你好81（2011）1565- 1578]。
The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo algorithm that is easy to implement but often suffers from slow convergence. The sandwich algorithm is an alternative that can converge much faster while requiring roughly the same computational effort per iteration. Theoretically, the sandwich algorithm always converges at least as fast as the corresponding DA algorithm in the sense that $\Vert {K^*}\Vert \le \Vert {K}\Vert$, where $K$ and $K^*$ are the Markov operators associated with the DA and sandwich algorithms, respectively, and $\Vert\cdot\Vert$ denotes operator norm. In this paper, a substantial refinement of this operator norm inequality is developed. In particular, under regularity conditions implying that $K$ is a trace-class operator, it is shown that $K^*$ is also a positive, trace-class operator, and that the spectrum of $K^*$ dominates that of $K$ in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. Furthermore, if the sandwich algorithm is constructed using a group action, as described by Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264--1274] and Hobert and Marchev [Ann. Statist. 36 (2008) 532--554], then there is strict inequality between at least one pair of eigenvalues. These results are applied to a new DA algorithm for Bayesian quantile regression introduced by Kozumi and Kobayashi [J. Stat. Comput. Simul. 81 (2011) 1565--1578].