  We propose and develop a novel and effective perfect sampling methodology for
simulating from posteriors corresponding to mixtures with either known (fixed)
or unknown number of components. For the latter we consider the Dirichlet
process-based mixture model developed by these authors, and show that our ideas
are applicable to conjugate, and importantly, to non-conjugate cases. As to be
expected, and, as we show, perfect sampling for mixtures with known number of
components can be achieved with much less effort with a simplified version of
our general methodology, whether or not conjugate or non-conjugate priors are
used. While no special assumption is necessary in the conjugate set-up for our
theory to work, we require the assumption of bounded parameter space in the
non-conjugate set-up. However, we argue, with appropriate analytical,
simulation, and real data studies as support, that such boundedness assumption
is not unrealistic and is not an impediment in practice. Not only do we
validate our ideas theoretically and with simulation studies, but we also
consider application of our proposal to three real data sets used by several
authors in the past in connection with mixture models. The results we achieved
in each of our experiments with either simulation study or real data
application, are quite encouraging.
