Suppose I have an experiment X with mutually exclusive outcomes from a set S. My goal is to determine the probability distribution for S. The problem is that I do not know how many elements are in S to begin with. I have an idea of an upper limit of the size of S (let's say 10) but I do not know exactly.
What is the Bayesian approach to determining the probability of each outcome? Is a Bayesian approach even appropriate here?
If a concrete example is needed, consider a bag with an infinite number of differently-colored marbles. I know that there are at most 10 unique marble colors, but I do not know how many. My desire is a Bayesian approach to estimate the number of colors in the bag and to determine the likelihood of drawing a marble of a given color.