  Estimation of stochastic processes evolving in a random environment is of
crucial importance for example to predict aircraft trajectories evolving in an
unknown atmosphere. For fixed parameter, interacting particle systems are a
convenient way to approximate such stochastic process. But the second level of
uncertainty provided by the environment parameters leads us to also consider
interacting particles on the parameter space. This novel algorithm is described
in this paper. It allows to approximate both a random environment and a
stochastic process evolving in this environment, given noisy observations of
the process. It is a sequential algorithm that generalizes island particle
models including a parameter. It is referred by us as labeled island particle
algorithm. We prove the convergence of the labeled island particle algorithm
and we establish $\mathbb{L}^p$ bound as well as time uniform $\mathbb{L}^p$
bound for the asymptotic error introduced by this double level of
approximation. Finally, we illustrate these results on a filtering problem
where one learns a dynamical parameter through noisy observations of a
stochastic process influenced by the parameter.
