  We present a novel method in the family of particle MCMC methods that we
refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the
existing PG with backward simulation (PG-BS) procedure, we use backward
sampling to (considerably) improve the mixing of the PG kernel. Instead of
using separate forward and backward sweeps as in PG-BS, however, we achieve the
same effect in a single forward sweep. We apply the PG-AS framework to the
challenging class of non-Markovian state-space models. We develop a truncation
strategy of these models that is applicable in principle to any
backward-simulation-based method, but which is particularly well suited to the
PG-AS framework. In particular, as we show in a simulation study, PG-AS can
yield an order-of-magnitude improved accuracy relative to PG-BS due to its
robustness to the truncation error. Several application examples are discussed,
including Rao-Blackwellized particle smoothing and inference in degenerate
state-space models.
