ABSTRACT
This chapter surveys the various forms of approximate Bayesian computation (ABC) algorithms that have been developed to sample from pABC. The earliest ABC samplers were basic rejection sampling algorithms. Improvements to general ABC samplers include increasing algorithmic efficiency by using quasi Monte Carlo methods, and the use of multi-level rejection sampling for variance reduction. Perhaps the biggest offshoot of ABC samplers is the more general pseudo-marginal Monte Carlo method, which implements exact Monte Carlo simulation with an unbiased estimate of the target distribution, of which ABC is a particular case. The idea of the marginal ABC sampler is closely related to the construction of the more recently developed pseudo-marginal sampler, a more general class of likelihood-free sampler that has gained popularity outside of the ABC setting. Markov chain Monte Carlo (MCMC) methods are a highly accessible class of algorithms for obtaining samples from complex distributions.
