  Sequential Monte Carlo (SMC) is a methodology for sampling approximately from
a sequence of probability distributions of increasing dimension and estimating
their normalizing constants. We propose here an alternative methodology named
Sequentially Interacting Markov Chain Monte Carlo (SIMCMC). SIMCMC methods work
by generating interacting non-Markovian sequences which behave asymptotically
like independent Metropolis-Hastings (MH) Markov chains with the desired
limiting distributions. Contrary to SMC, SIMCMC allows us to iteratively
improve our estimates in an MCMC-like fashion. We establish convergence results
under realistic verifiable assumptions and demonstrate its performance on
several examples arising in Bayesian time series analysis.
