We discuss relationships between the existing methods for MCMC exploration of model spaces, including the reversible jump sampler of Green (1995),
the `model composition' approach of Carlin and Chib (1995), the MC$^3$
techniques of Madigan and Raftery (1995) and MCMC methods for variable
selection such as George and McCulloch (1993), Kuo and Mallick (1997)
and Geweke (1996). We link these different methods together
through a composite model space similar to that used by Carlin and Chib in which a model of constant dimensionality
is created by considering the product space of parameters from all
possible models within the candidate set and the model indexing
variable. In the examples given in their paper, Carlin and Chib apply a straightforward Gibbs
sampler to the composite space which renders the method impracticable for comparison between more than a small handful
of models. We show that the other methods
of MCMC model selection can be obtained by applying different forms of MCMC sampling to the composite space.
The results shed some light upon the issues of `pseudo-prior' selection in the case
of the Carlin and Chib sampler and choice of proposal distribution in the case
 of Green's reversible jump method. Furthermore, we propose efficient reversible jump proposal schemes which
take advantage of any analytic structure that may be present in the
model. The method is compared with a standard reversible jump scheme
for the problem of model order uncertainty in  autoregressive time
series.