Assume that we have an unfair die, and our task is to determine the probabilities of rolling a 1, a 2, a 3 and so on by rolling the die. Unfortunately we have no way of knowing how many sides the die has, only that it has finitely many sides.
Suppose that we roll the die 1000 times, and every number from 1 to 7 gets rolled, some very often, some only once, as the die is extremely unfair. We can calculate the multinomial probabilities trivially, but trying to quantify the uncertainty associated with those probabilities is much harder.
Literature methods for simultaneous multinomial confidence intervals always seem to assume that there is no uncertainty in the number of categories, but that assumption is not met here: if we were to roll say, 100 000 times instead, we might get a couple rolls that land on a number higher than 7. In fact with a finite number of rolls, the best we can hope for is an upper bound on the probability of unobserved categories.
Is there a known way to account for the possibility of unobserved events, and thus an uncertainty in the number of categories, when calculating multinomial confidence intervals?