I see, so you talk about the case where we not observe any events at all? In that case I would indeed be surprised if it was possible to (mathematically) prove

I see, so you talk about the case where we not observe any events at all? In that case I would indeed be surprised if it was possible to (mathematically) prove a theory using this negative knowledge. We can make use of negative knowledge to draw conclusions about a given system though. For example, if we do not know whether an electron bound to an atom is in its ground state or not we might observe the system to see if it emits a photon or not. The longer we observe the system not emitting a photon, the more likely it is that it is in its ground state. So here we actually gain quantifiable knowledge about a system from the observation of a "non-event".