This provides an efficient method for drawing samples from BetaBinomial(n,alpha,beta) when alpha and beta are positive integers. What about when alpha and beta are positive reals, but not integers?
[edit]
What about using two measuring sticks instead of red-coloured and blue-coloured balls? If you pick from the red measuring stick, increase its length by one; and vice versa for the blue one. The lengths of the sticks can be varied continuously. Does this allow for alpha and beta to be arbitrary positive reals?
I love this. I'm not sure why, but I had never made the connection to sampling with and without replacement, and sampling/measurement effects on future samples/measurements. It had just never occurred to me before.
[edit]
What about using two measuring sticks instead of red-coloured and blue-coloured balls? If you pick from the red measuring stick, increase its length by one; and vice versa for the blue one. The lengths of the sticks can be varied continuously. Does this allow for alpha and beta to be arbitrary positive reals?