In a stochastic noise setting the Lepskij balancing principle for choosing
the regularization parameter in the regularization of inverse problems is
depending on a parameter $\tau$ which in the currently known proofs is
depending on the unknown noise level of the input data. However, in practice
this parameter seems to be obsolete.

We will present an explanation for this behavior by using a stochastic model
for noise and initial data. Furthermore, we will prove that a small
modification of the algorithm also improves the performance of the method, in
both speed and accuracy.