Among the computationally intensive methods for fitting complex multilevel models, the Gibbs sampler is especially popular owing to its simplicity and power to effectively generate samples from a high-dimensional probability distribution. The Gibbs sampler, however, is often justifiably criticized for its sometimes slow convergence, especially when it is used to fit highly structured complex models. The recently proposed Partially Collapsed Gibbs (PCG) sampler offers a new strategy for

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... the convergence characteristics of a Gibbs sampler. A PCG sampler achieves faster convergence by reducing the conditioning in some or all of the component draws of its parent Gibbs sampler. Although this strategy can significantly improve convergence, it must be implemented with care to be sure that the desired stationary distribution is preserved. In some cases the set of conditional distributions sampled in a PCG sampler may be functionally incompatible and permuting the order of draws can change the stationary distribution of the chain. In this article, we draw an analogy between the PCG sampler and certain efficient EM-type algorithms that helps to explain the computational advantage of PCG samplers and to suggest when they might be used in practice. We go on to illustrate the PCG samplers in three substantial examples drawn from our applied work: a multilevel spectral model commonly used in high-energy astrophysics, a piecewise-constant multivariate time series model, and a joint imputation model for nonnested data. These are all useful highly structured models that involve computational challenges that can be solved using PCG samplers. The examples illustrate not only the computation advantage of PCG samplers but also how they should be constructed to maintain the desired stationary distribution. Supplemental materials for the examples given in this article are available online.