  We generalize the heuristic parameter choice rule of Hanke-Raus for quadratic
regularization to general variational regularization for solving linear as well
as nonlinear ill-posed inverse problems in Banach spaces. Under source
conditions formulated as variational inequalities, we obtain a posteriori error
estimates in term of Bregman distance. By imposing certain conditions on the
random noise, we establish four convergence results; one relies on the source
conditions and the other three do not depend on any source conditions.
Numerical results are presented to illustrate the performance.
