Barker's algorithm for Bayesian inference with intractable likelihoods

In this expository paper we abstract and describe a simple MCMC scheme for sampling from intractable target densities. The approach has been introduced in Gon\c{c}alves et al. (2017a) in the specific context of jump-diffusions, and is based on the Barker's algorithm paired with a simple Bernoulli factory type scheme, the so called 2-coin algorithm. In many settings it is an alternative to standard Metropolis-Hastings pseudo-marginal method for simulating from intractable target densities. Although Barker's is well-known to be slightly less efficient than Metropolis-Hastings, the key advantage of our approach is that it allows to implement the "marginal Barker's" instead of the extended state space pseudo-marginal Metropolis-Hastings, owing to the special form of the accept/reject probability. We shall illustrate our methodology in the context of Bayesian inference for discretely observed Wright-Fisher family of diffusions.