A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms
A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms
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DOI：
10.1214/009053607000000569
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发表时间：
2008-04-01
影响因子：
4.5
通讯作者：
Marchev, Dobrin
中科院分区：
文献类型：
--
作者：
Hobert, James P.;Marchev, Dobrin
数据扩充（DA）算法是广泛使用的马尔可夫链蒙特卡罗（MCMC）算法，其基于形式为p（x垂直条x '）=积分y fx垂直条y（x垂直条y）fY垂直条X（y垂直条x'）dy的马尔可夫转移密度，其中fX垂直条Y和fY垂直条X是条件密度。Liu和Wu的PX-DA和边际增强算法[J. Amer. Statistic. Assoc.94（1999）1264-1274]以及Meng和货车Dyk [Biometrika 86（1999）301-320]是DA的替代方案，它们通常收敛得快得多，并且仅在计算上要求稍微高一些。这些替代算法的转移密度可以写成以下形式：PR（x垂直条x '）= integral Y integral y fX垂直条Y（x垂直条y'）R（y，dy '）fY垂直条X（y垂直条x'）dy，其中R是Y上的马尔可夫转移函数。证明了当R满足一定条件时，PR驱动的MCMC算法在中心极限定理和算子范数意义下的性能至少与p驱动的MCMC算法一样好.这些结果带来的DA，PX-DA和边缘增强算法的理论比较。我们的重点是刘和吴所利用的群体结构可用的情况。我们证明了基于Haar测度的PX-DA算法至少与任何使用适当的先验构造的PX-DA算法一样好。
The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo (MCMC) algorithm that is based on a Markov transition density of the form p(x vertical bar x') = integral y fx vertical bar y (x vertical bar y)fY vertical bar X (y vertical bar x') dy, where fX vertical bar Y and fY vertical bar X are conditional densities. The PX-DA and marginal augmentation algorithms of Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264-1274] and Meng and van Dyk [Biometrika 86 (1999) 301-320] are alternatives to DA that often converge much faster and are only slightly more computationally demanding. The transition densities of these alternative algorithms can be written in the form PR (x vertical bar x') = integral Y integral y fX vertical bar Y (x vertical bar y') R(y, dy')fY vertical bar X (y vertical bar x') dy, where R is a Markov transition function on Y. We prove that when R satisfies certain conditions, the MCMC algorithm driven by PR is at least as good as that driven by p in terms of performance in the central limit theorem and in the operator norm sense. These results are brought to bear on a theoretical comparison of the DA, PX-DA and marginal augmentation algorithms. Our focus is on situations where the group structure exploited by Liu and Wu is available. We show that the PX-DA algorithm based on Haar measure is at least as good as any PX-DA algorithm constructed using a proper prior on the group.