Thank God That Regressing Y on X is Not the Same as Regressing X on Y: Direct and Indirect Residual Augmentations

What does regressing Y on X versus regressing X on Y have to do with Markov chain Monte Carlo (MCMC)? It turns out that many strategies for speeding up data augmentation (DA) type algorithms can be understood as fostering independence or “de-correlation” between a regression function and the corresponding residual, thereby reducing or even eliminating dependence among MCMC iterates. There are two general classes of algorithms, those corresponding to regressing parameters on augmented data/auxiliary variables and those that operate the other way around. The interweaving strategy of Yu and Meng provides a general recipe to automatically take advantage of both, and it is the existence of two different types of residuals that makes the interweaving strategy seemingly magical in some cases and promising in general. The concept of residuals—which depends on actual data—also highlights the potential for substantial improvements when DA schemes are allowed to depend on the observed data. At the same time, there is an intriguing phase transition type of phenomenon regarding choosing (partially) residual augmentation schemes, reminding us once more of the prevailing issue of trade-off between robustness and efficiency. This article reports on these latest theoretical investigations (using a class of normal/independence models) and empirical findings (using a posterior sampling for a probit regression) in the search for effective residual augmentations—and ultimately more MCMC algorithms—that meet the 3-S criterion: simple, stable, and speedy. Supplementary materials for the article are available online.