  We propose a general framework for studying optimal impulse control problem
in the presence of uncertainty on the parameters. Given a prior on the
distribution of the unknown parameters, we explain how it should evolve
according to the classical Bayesian rule after each impulse. Taking these
progressive prior-adjustments into account, we characterize the optimal policy
through a quasi-variational parabolic equation, which can be solved
numerically. The derivation of the dynamic programming equation seems to be new
in this context. The main difficulty lies in the nature of the set of controls
which depends in a non trivial way on the initial data through the filtration
itself.
