  The particle-based, rapid incremental smoother (PARIS) is a sequential Monte
Carlo technique allowing for efficient online approximation of expectations of
additive functionals under Feynman--Kac path distributions. Under weak
assumptions, the algorithm has linear computational complexity and limited
memory requirements. It also comes with a number of non-asymptotic bounds and
convergence results. However, being based on self-normalised importance
sampling, the PARIS estimator is biased; its bias is inversely proportional to
the number of particles but has been found to grow linearly with the time
horizon under appropriate mixing conditions. In this work, we propose the
Parisian particle Gibbs (PPG) sampler, whose complexity is essentially the same
as that of the PARIS and which significantly reduces the bias for a given
computational complexity at the price of a modest increase in the variance.
This method is a wrapper in the sense that it uses the PARIS algorithm in the
inner loop of particle Gibbs to form a bias-reduced version of the targeted
quantities. We substantiate the PPG algorithm with theoretical results,
including new bounds on bias and variance as well as deviation inequalities. We
illustrate our theoretical results with numerical experiments supporting our
claims.
