  Many exact Markov chain Monte Carlo algorithms have been developed for
posterior inference in Bayesian nonparametric models which involve
infinite-dimensional priors. However, these methods are not generic and special
methodology must be developed for different classes of prior or different
models. Alternatively, the infinite-dimensional prior can be truncated and
standard Markov chain Monte Carlo methods used for inference. However, the
error in approximating the infinite-dimensional posterior can be hard to
control for many models. This paper describes an adaptive truncation method
which allows the level of the truncation to be decided by the algorithm and so
can avoid large errors in approximating the posterior. A sequence of truncated
priors is constructed which are sampled using Markov chain Monte Carlo methods
embedded in a sequential Monte Carlo algorithm. Implementational details for
infinite mixture models with stick-breaking priors and normalized random
measures with independent increments priors are discussed. The methodology is
illustrated on infinite mixture models, a semiparametric linear mixed model and
a nonparametric time series model.
