  In this note we solve a general statistical inverse problem under absence of
knowledge of both the noise level and the noise distribution via application of
the (modified) heuristic discrepancy principle. Hereby the unbounded
(non-Gaussian) noise is controlled via introducing an auxiliary discretisation
dimension and choosing it in an adaptive fashion. We first show convergence for
completely arbitrary compact forward operator and ground solution. Then the
uncertainty of reaching the optimal convergence rate is quantified in a
specific Bayesian-like environment.
