We consider a version of the data augmentation algorithm of Tanner and Wong, which is a special case of the Gibbs sampler. Using ideas from Harris recurrence, we derive quantitative, a priori bounds on the number of iterations required to achieve convergence. Our analysis involves relating the Markov chain to an associated dynamical system.
"Rates of Convergence for Data Augmentation on Finite Sample Spaces." Ann. Appl. Probab. 3 (3) 819 - 839, August, 1993. https://doi.org/10.1214/aoap/1177005366