  Consider a finite sample from an unknown distribution over a countable
alphabet. Unobserved events are alphabet symbols which do not appear in the
sample. Estimating the probabilities of unobserved events is a basic problem in
statistics and related fields, which was extensively studied in the context of
point estimation. In this work we introduce a novel interval estimation scheme
for unobserved events. Our proposed framework applies selective inference, as
we construct confidence intervals (CIs) for the desired set of parameters.
Interestingly, we show that obtained CIs are dimension-free, as they do not
grow with the alphabet size. Further, we show that these CIs are (almost)
tight, in the sense that they cannot be further improved without violating the
prescribed coverage rate. We demonstrate the performance of our proposed scheme
in synthetic and real-world experiments, showing a significant improvement over
the alternatives. Finally, we apply our proposed scheme to large alphabet
modeling. We introduce a novel simultaneous CI scheme for large alphabet
distributions which outperforms currently known methods while maintaining the
prescribed coverage rate.
