Efficient Bernoulli factory MCMC for intractable posteriors

Accept-reject based Markov chain Monte Carlo (MCMC) algorithms have traditionally utilised acceptance functions that can be explicitly written as a function of the ratio of the target density at the two contested points. This feature is rendered almost useless in Bayesian posteriors with unknown functional forms. We introduce a new family of MCMC acceptance probabilities that has the distinguishing feature of not being a function of the ratio of the target density at the two points. We present two efficient and stable Bernoulli factories that generate events within this class of acceptance probabilities. The resulting portkey Barker's algorithms are exact and computationally more efficient that the current state-of-the-art.