  The particle Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm to
sample from the full posterior distribution of a state-space model. It does so
by executing Gibbs sampling steps on an extended target distribution defined on
the space of the auxiliary variables generated by an interacting particle
system. This paper makes the following contributions to the theoretical study
of this algorithm. Firstly, we present a coupling construction between two
particle Gibbs updates from different starting points and we show that the
coupling probability may be made arbitrarily close to one by increasing the
number of particles. We obtain as a direct corollary that the particle Gibbs
kernel is uniformly ergodic. Secondly, we show how the inclusion of an
additional Gibbs sampling step that reselects the ancestors of the particle
Gibbs' extended target distribution, which is a popular approach in practice to
improve mixing, does indeed yield a theoretically more efficient algorithm as
measured by the asymptotic variance. Thirdly, we extend particle Gibbs to work
with lower variance resampling schemes. A detailed numerical study is provided
to demonstrate the efficiency of particle Gibbs and the proposed variants.
