标题： 数据增强、边际增强和PX-DA算法的理论比较 显示英文标题

标题： A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms

摘要： 数据增强（Data Augmentation, DA）算法是一种广泛应用的马尔可夫链蒙特卡洛（Markov Chain Monte Carlo, MCMC）算法，其基于一种形式为$p(x|x')=\int_{\mathsf{Y}}f_{X|Y}(x|y)f_{Y|X}(y|x') dy$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 [J. Amer. Statist. Assoc. 94 (1999) 1264--1274] 提出的 PX-DA 算法以及 Meng 和 van Dyk [Biometrika 86 (1999) 301--320] 提出的边际增强算法是 DA 算法的替代方法，通常收敛速度更快，且仅略微增加计算负担。 这些替代算法的转移密度可以写成$p_R(x|x')=\int_{\mathsf{Y}}\int _{\mathsf{Y}}f_{X|Y}(x|y')R(y,dy')f_{Y|X}(y|x') dy$的形式，其中$R$是$\mathsf{Y}$上的一个马尔可夫转移函数。 我们证明了当$R$满足某些条件时，由$p_R$驱动的 MCMC 算法在中心极限定理和算子范数意义下至少与由$p$驱动的算法一样好。 这些结果被用于 DA、PX-DA 和边际增强算法的理论比较。 我们的重点是 Liu 和 Wu 所利用的群结构可用的情况。 我们表明，基于 Haar 测度的 PX-DA 算法至少与任何使用群上的适当先验构造的 PX-DA 算法一样好。 显示英文摘要

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.