  This paper shows that one cannot learn the probability of rare events without
imposing further structural assumptions. The event of interest is that of
obtaining an outcome outside the coverage of an i.i.d. sample from a discrete
distribution. The probability of this event is referred to as the "missing
mass". The impossibility result can then be stated as: the missing mass is not
distribution-free PAC-learnable in relative error. The proof is
semi-constructive and relies on a coupling argument using a dithered geometric
distribution. This result formalizes the folklore that in order to predict rare
events, one necessarily needs distributions with "heavy tails".
