  In this paper we study asymptotic properties of different
data-augmentation-type Markov chain Monte Carlo algorithms sampling from
mixture models comprising discrete as well as continuous random variables. Of
particular interest to us is the situation where sampling from the conditional
distribution of the continuous component given the discrete component is
infeasible. In this context, we cast Carlin & Chib's pseudo-prior method into
the framework of mixture models and discuss and compare different variants of
this scheme. We propose a novel algorithm, the FCC sampler, which is less
computationally demanding than any Metropolised Carlin & Chib-type algorithm.
The significant gain of computational efficiency is however obtained at the
cost of some asymptotic variance. The performance of the algorithm vis-\`a-vis
alternative schemes is investigated theoretically, using some recent results
obtained in [3] for inhomogeneous Markov chains evolving alternatingly
according to two different reversible Markov transition kernels, as well as
numerically.
