I have a problem that seems tractable, which I will attempt to describe with classic balls and buckets probability nomenclature.
Imagine someone brings you a bag of colored balls. This sample was selected at random from one of many large buckets of balls. Each bucket contains 1,000,000 balls, and each ball is one of 1,000 colors. For any bucket, the proportion of balls for each color is fixed, and we know what it is. For example, Bucket A always has 0.02 Red, 0.013 Green, 0.009 Blue, etc.
So--how would I go about estimating which bucket the sample of balls came from? I know that if I have a bag (sample) of 1,000,000 balls, I can say unambiguously which bucket that came from, as the proportions are defined. However, what do I do if I have only 2000 balls in the sample? It seems to me I should be able to calculate quite specifically the likelihood that the sample came from a particular bucket.
Any help would be greatly appreciated! I'll be doing this in code (likely Python).
Best regards, Dan