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?

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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 went to this really nice talk by Ghahramani on Bayesian Nonparametrics at 2011 NIPS. Worth seeing polya urn model in action.

http://mlg.eng.cam.ac.uk/zoubin/

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.