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Consider a randomized algorithm that draws samples exactly from adistribution using recursion. Such an algorithm is called a perfect simulation,and here a variety of methods for building this type of algorithm are shown toderive from the same result: the Fundamental Theorem of Perfect Simulation(FTPS). The FTPS gives two necessary and sufficient conditions for the outputof a recursive probabilistic algorithm to come exactly from the desireddistribution. First, the algorithm must terminate with probability 1. Second,the algorithm must be locally correct, which means that if the recursive callsin the original algorithm are replaced by oracles that draw from the desireddistribution, then this new algorithm can be proven to be correct. While it isusually straightforward to verify these conditions, they are surprisinglypowerful, giving the correctness of Acceptance/Rejection, Coupling from thePast, the Randomness Recycler, Read-once CFTP, Partial Rejection Sampling,Partially Recursive Acceptance Rejection, and various Bernoulli Factories. Weillustrate the use of this algorithm by building a new Bernoulli Factory forlinear functions that is 41\% faster than the previous method.

Designing Perfect Simulation Algorithms using Local Correctness