  This article provides a new theory for the analysis of forward and backward
particle approximations of Feynman-Kac models. Such formulae are found in a
wide variety of applications and their numerical (particle) approximation are
required due to their intractability. Under mild assumptions, we provide sharp
and non-asymptotic first order expansions of these particle methods,
potentially on path space and for possibly unbounded functions. These
expansions allows one to consider upper and lower bound bias type estimates for
a given time horizon $n$ and particle number $N$; these non-asymptotic
estimates are of order $\mathcal{O}(n/N)$. Our approach is extended to tensor
products of particle density profiles, leading to new sharp and non-asymptotic
propagation of chaos estimates. The resulting upper and lower bound propagation
of chaos estimates seems to be the first result of this kind for mean field
particle models. As a by-product of our results, we also provide some analysis
of the particle Gibbs sampler, providing first order expansions of the kernel
and minorization estimates.
