Approximate Bayesian computation (ABC) algorithms are a class of Monte Carlo methods for doing inference when the likelihood function can be simulated from, but not explicitly evaluated. This situation commonly occurs when using even relatively simple stochastic models. The algorithms can be viewed as methods for combining the scientific knowledge encoded in a computer model, with the empirical information contained in the data. The methods have become popular in the biological sciences, particularly in fields such as genetics and systematic biology, as they are simple to apply, and can be used on nearly any problem.
However, there are several problems with ABC algorithms: they can be inefficient if applied naively; they only give approximate answers with the quality of the approximation unknown; they rely on a vector of summary statistics that is difficult to choose. In the first part of this tutorial, I will introduce the basic ideas behind ABC algorithms and illustrate their use on a problem from climate science. In the second part, I will describe some of the recent advances in ABC research, including regression adjustment methods, automatic summary selection, and the use of generalized acceptance kernels.