  In this article we consider Bayesian parameter inference associated to
partially-observed stochastic processes that start from a set B0 and are
stopped or killed at the first hitting time of a known set A. Such processes
occur naturally within the context of a wide variety of applications. The
associated posterior distributions are highly complex and posterior parameter
inference requires the use of advanced Markov chain Monte Carlo (MCMC)
techniques. Our approach uses a recently introduced simulation methodology,
particle Markov chain Monte Carlo (PMCMC) (Andrieu et. al. 2010 [1]), where
sequential Monte Carlo (SMC) approximations (see Doucet et. al. 2001 [18] and
Liu 2001 [27]) are embedded within MCMC. However, when the parameter of
interest is fixed, standard SMC algorithms are not always appropriate for many
stopped processes. In Chen et. al. [11] and Del Moral 2004 [15] the authors
introduce SMC approximations of multi-level Feynman-Kac formulae, which can
lead to more efficient algorithms. This is achieved by devising a sequence of
nested sets from B0 to A and then perform the resampling step only when the
samples of the process reach intermediate level sets in the sequence.
Naturally, the choice of the intermediate level sets is critical to the
performance of such a scheme. In this paper, we demonstrate that multi-level
SMC algorithms can be used as a proposal in PMCMC. In addition, we propose a
flexible strategy that adapts the level sets for different parameter proposals.
Our methodology is illustrated on the coalescent model with migration.
