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

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?