  This paper introduces methodology for performing Bayesian inference
sequentially on a sequence of posteriors on spaces of different dimensions. We
show how this may be achieved through the use of sequential Monte Carlo (SMC)
samplers (Del Moral et al., 2006, 2007), making use of the full flexibility of
this framework in order that the method is computationally efficient. In
particular, we introduce the innovation of using deterministic transformations
to move particles effectively between target distributions with different
dimensions. This approach, combined with adaptive methods, yields an extremely
flexible and general algorithm for Bayesian model comparison that is suitable
for use in applications where the acceptance rate in reversible jump Markov
chain Monte Carlo (RJMCMC) is low. We demonstrate this approach on the
well-studied problem of model comparison for mixture models, and for the novel
application of inferring coalescent trees sequentially, as data arrives.
