Suppose we have many identical boxes, too many that we cannot open all of them.
Each box has finitely many balls. The number of balls in each box can be very large, but is finite. Balls are identical but different in color.
The colors include red, white, blue, black, yellow, etc. We do not know the total number of colors (finite) in advance. Each color can be tagged as dark, light and intermediate. Although we do not know how many colors are there in advance, given a new color, a fixed oracle can classify it into one of {dark, light, intermediate}.
To summary: a box has several balls, each ball has a color, and each color can be classified as one of {dark, light, intermediate}.
The QUESTION is: how to SAMPLE from the population of boxes, and to ESTIMATE ("the total number of colors, as well as" DELETE, see UPDATE below) the proportion of dark, light and intermediate?
(UPDATE: it's unlikely to estimate the number of colors since in this problem, if we don't open all the boxes, many colors will be missed, and we have no extra information to know that.)
It would be ideal if someone can refer some textbooks or papers on this problem to me.
Example & explanation: 3 boxes arranged as {red,blue,white,black}, {red,blue,yellow}, {red}.
Suppose the oracle tells us that the colors are classified as {dark: black}, {light: white, yellow}, {intermediate: red, blue}
The total number of colors is 5, and the dark-light-intermediate ratio is 0.2:0.4:0.4. Although there are one white ball and one yellow ball while three red balls and two blue balls, they contribute equal in terms of distinct colors, and hence the proportion of light/intermediate.
PS: I think this problem is more related to combinatorics than statistics, and hence post it here instead of Cross-Validate website.