The Data Augmentation (DA) algorithm is frequently used in posterior sampling, especially when the model has a hierarchical structure containing latent variables. Marginal Data Augmentation (MDA) and Ancillarity-Sufficiency Interweaving Strategy (ASIS) are both effective tools to improve the convergence of DA. MDA introduces a working parameter α in the augmented data model that is not identifiable under the observed data model. Although the resulting joint chain may not be positive recurrent, the marginal chain for the parameter can mix very efficiently. ASIS considers a pair of special DA schemes: the so-called sufficient and ancillary schemes. If the sampler corresponding to one of these is fast, the sampler corresponding to the other is typically slow. The ASIS strategy can often substantially outperform both of the parent DA samplers. We propose to combine these strategies to further improve efficiency. We apply this combining strategy on a factor analysis model to show its effect. The intuition of combining MDA and ASIS in this factor analysis model is that MDA is efficient in improving the convergence of the variance parameter, ∑, but has very little effect on the factor loading matrix, β. ASIS can result in samplers with much better convergence properties, but ancillary and sufficient augmentations can be difficult or impossible to derive in complex models unless we condition on a subset of the parameters. Thus, we consider handling ∑ with MDA, and deriving ASIS for β given Σ. By combining MDA and ASIS in this way, we improve the convergence of both ∑ and β significantly. Furthermore, we can guarantee the subchain {(β^{(t)},∑^{(t)}),t≥0} resulting from the combining strategy converges to the target distribution, i.e., p(β,∑|Y).

*Joint work with David van Dyk (Imperial College London).*