  This paper considers the problem of using MCMC to fit sparse Bayesian models
based on normal scale-mixture priors. Examples of this framework include the
Bayesian LASSO and the horseshoe prior. We study the usefulness of parameter
expansion (PX) for improving convergence in such models, which is notoriously
slow when the global variance component is near zero. Our conclusion is that
parameter expansion does improve matters in LASSO-type models, but only
modestly. In most cases this improvement, while noticeable, is less than what
might be expected, especially compared to the improvements that PX makes
possible for models very similar to those considered here. We give some
examples, and we attempt to provide some intuition as to why this is so. We
also describe how slice sampling may be used to update the global variance
component. In practice, this approach seems to perform almost as well as
parameter expansion. As a practical matter, however, it is perhaps best viewed
not as a replacement for PX, but as a tool for expanding the class of models to
which PX is applicable.
