Seeking efficient data augmentation schemes via conditional and marginal augmentation

Data augmentation, sometimes known as the method of auxiliary variables, is a powerful tool for constructing optimisation and simulation algorithms. In the context of optimisation, Meng & van Dyk (1997, 1998) reported several successes of the 'working parameter' approach for constructing efficient data-augmentation schemes for fast and simple EM-type algorithms. This paper investigates the use of working parameters in the context of Markov chain Monte Carlo, in particular in the context of Tanner & Wong's (1987) data augmentation algorithm, via a theoretical study of two working-parameter approaches, the conditional augmentation approach and the marginal augmentation approach. Posterior sampling under the univariate t model is used as a running example, which particularly illustrates how the marginal augmentation approach obtains a fast-mixing positive recurrent Markov chain by first constructing a nonpositive recurrent Markov chain in a larger space.