In recent years, a variety of extensions and refinements have been developed for data augmentation (DA) based model fitting routines. These developments aim to extend the application, improve the speed, and/or simplify the implementation of DA methods, such as the deterministic EM algorithm for mode finding and stochastic methods including the DA algorithm and the method of auxiliary variable for posterior sampling. In this paper we graphically illustrate and compare a number of these extensions all of which aim to maintain the simplicity and computation stability of their predecessors. We also show the applicability of DA methods for handling complex models with highly hierarchical structure, using a high-energy high-resolution spectral imaging model for data from a new generation of satellite telescopes, such as the soon to be launched Chandra space observatory.
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