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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.

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You can solve this problem with a hierarchical model where the number of colors in the bag is first drawn from a distribution over integers ranging from 1 to infinity (say, a Poisson distribution), then a subset of colors are drawn at random from the possible colors, and finally a probability vector over these colors is chosen at random. For details, see the paper "Efficient Bayesian Parameter Estimation in Large Discrete Domains".

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