  The Partially Collapsed Gibbs (PCG) sampler offers a new strategy for
improving the convergence of a Gibbs sampler. PCG achieves faster convergence
by reducing the conditioning in some of the draws of its parent Gibbs sampler.
Although this can significantly improve convergence, care must be taken to
ensure that the stationary distribution is preserved. The conditional
distributions sampled in a PCG sampler may be incompatible and permuting their
order may upset the stationary distribution of the chain. Extra care must be
taken when Metropolis-Hastings (MH) updates are used in some or all of the
updates. Reducing the conditioning in an MH within Gibbs sampler can change the
stationary distribution, even when the PCG sampler would work perfectly if MH
were not used. In fact, a number of samplers of this sort that have been
advocated in the literature do not actually have the target stationary
distributions. In this article, we illustrate the challenges that may arise
when using MH within a PCG sampler and develop a general strategy for using
such updates while maintaining the desired stationary distribution. Theoretical
arguments provide guidance when choosing between different MH within PCG
sampling schemes. Finally we illustrate the MH within PCG sampler and its
computational advantage using several examples from our applied work.
