Cognitive radio (CR) networks can re-use the RF spectrum licensed to the primary user (PU) network by carefully controlling the interference to the PUs. However, due to lack of explicit support from the PU system, CR sensing algorithms often face difficulty in acquiring CR-to-PU channels accurately. Moreover, the sensing algorithms cannot detect silent PU receivers, which nevertheless have to be protected. In order to achieve aggressive spectrum re-use even in such challenging scenarios, a CR power control problem with probabilistic interference constraints is formulated. Both log-normal shadowing and small-scale fading uncertainties are taken into account through suitable approximations. In particular, a weighted sum-rate maximization problem is considered, whose Karush-Kuhn-Tucker points are obtained via sequential geometric programming. Numerical tests verify the performance of our novel approach.