How can Bayes modeling be used to solve the known statistical urn and ball problem?
There is one urn. There are multiple colored balls within the same urn. There are five colors (Blue,Yellow,Green,Black and White).No data with regards to how many of each color.
If a sample is taken from the urn, what methods (regarding Bayes etc.) can be used to establish an overall estimation of the population in the urn?
My main goal is to graphically present the population distribution after a sample has been taken.
Any knowledge regarding bayesian updating after the sampling process will be appreciated.
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I suppose you know what the total population is.
Because there’s no data about how many of each color there are you cannot derive an a priori distribution function. However, you do know that there are five colors type of balls. Then I think you can construct a “mixture distribution” after the sample. There are 2 ways:
1) Assume as an a priori distribution, a uniform distribution between the colors (you can’t assume anymore because the absence of data, but at least you’re giving a way to ponder that there are 5 colors and not less , even if in your sample you only take out balls of less than 5 colors). Then do the sample, and calculate the a posteriori distribution.
2) Do the sample:
a) If you have 5 colors in your sample, take the sample distribution as the desired approximation distribution function.
b) If you have less than 5 colors in you sample, define the uniform distribution between the colors as your a priori distribution , and then with the sample distribution you have already obtained, calculate the a posteriori distribution.
I cannot justify why, but i think the second approach is better.