数据增强、边缘增强和PX-DA算法的理论比较
A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms
摘要：
数据增强（DA）算法是一种广泛使用的马尔可夫链蒙特卡罗（MCMC）算法，其基于$p(x|x')=\int_{\mathsf{Y}}f_{X|Y}(x|y)f_{Y|X}(y|x') dy$形式的马尔可夫转移密度，其中$f_{X|Y}$和$f_{Y|X}$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van Dyk [Biometrika 86（1999）301- 320]是DA的替代方案，它们通常收敛得更快，并且仅在计算上要求稍高。这些替代算法的转换密度可以写成$p_R(x|x')=\int_{\mathsf{Y}}\int _{\mathsf{Y}}f_{X|Y}(x|y')R(y,dy')f_{Y|X}(y|x') dy$的形式，其中$R$是$\mathsf{Y}$上的马尔可夫转换函数。证明了当$R$满足一定条件时，由$p_R$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由$p$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。
英文摘要：
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 $p(x|x')=\int_{\mathsf{Y}}f_{X|Y}(x|y)f_{Y|X}(y|x') dy$, where $f_{X|Y}$ and $f_{Y|X}$ 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 $p_R(x|x')=\int_{\mathsf{Y}}\int _{\mathsf{Y}}f_{X|Y}(x|y')R(y,dy')f_{Y|X}(y|x') dy$, where $R$ is a Markov transition function on $\mathsf{Y}$. We prove that when $R$ satisfies certain conditions, the MCMC algorithm driven by $p_R$ 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.
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