ABSTRACT
Approximate Bayesian computation (ABC) is a popular method that consists of defining an alternative likelihood function, which is also in general intractable, but naturally lends itself to pseudo-marginal computations, hence, making the approach of practical interest. This chapter shows the connections of ABC Markov chain Monte Carlo (MCMC) with pseudo-marginal algorithms, reviews their existing theoretical results, and discusses how these can inform practice and hopefully lead to fruitful methodological developments. It describes standard performance measures for MCMC algorithms and a summary of some known theoretical results relating the properties of to the performance of pseudo-marginal algorithms. The chapter considers the comparing different variations of the noisy algorithm. It presents a relevant subset of theory and directions in methodological research pertaining to ABC-MCMC algorithms. The chapter suggests that when the model admits specific structure, alternatives to the simple ABC method presented here may be more computationally efficient.
