## A discrete Bernoulli factory

**A** rather confusing (and now closed) question on X validated contained an interesting challenge of simulating an arbitrary discrete distribution using a single (standard) dice. It indeed made me think of the (more challenging) Bernoulli factory problem of simulating B(f(p)) using a B(p) simulator (with p unknown). I still do not see what the optimal solution is but the core challenge is to avoid simulating U(0,1) variate by exploiting the discrete nature of the target. Which may be an issue if the probabilities of the target are irrational and one is considering the cdf inversion approach. An alternative is to use an accept-reject approach, which also works for discrete distributions, by first deriving an instrumental distribution on the discrete support of the target from dice rolls, second finding the maximum of the ratio instrument to target, and third devising a discrete approach to selecting a generation with a probability taking a finite number of values. Which may prove quite costly. Finally, the least debatable approach is to turn the dice into a Uniform generator by using each draw as a digit in the base 5 representation of this Uniform variate, up to the precision desired for the resolution, and then apply the most efficient algorithm for the target distribution.

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