  Accept-reject based Markov chain Monte Carlo (MCMC) algorithms have
traditionally utilised acceptance probabilities 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 stable Bernoulli factories
that generate events within this class of acceptance probabilities. The
efficiency of our methods rely on obtaining reasonable local upper or lower
bounds on the target density and we present two classes of problems where such
bounds are viable: Bayesian inference for diffusions and MCMC on constrained
spaces. The resulting portkey Barker's algorithms are exact and computationally
more efficient that the current state-of-the-art.
