Computing Bayes: Bayesian Computation from 1763 to the 21st Century

The Bayesian statistical paradigm uses the language of probability to express uncertainty about the phenomena that generate observed data. Probability distributions thus characterize Bayesian inference, with the rules of probability used to transform prior probability distributions for all unknowns - models, parameters, latent variables - into posterior distributions, subsequent to the observation of data. Conducting Bayesian inference requires the evaluation of integrals in which these probability distributions appear. Bayesian computation is all about evaluating such integrals in the typical case where no analytical solution exists. This paper takes the reader on a chronological tour of Bayesian computation over the past two and a half centuries. Beginning with the one-dimensional integral first confronted by Bayes in 1763, through to recent problems in which the unknowns number in the millions, we place all computational problems into a common framework, and describe all computational methods using a common notation. The aim is to help new researchers in particular - and more generally those interested in adopting a Bayesian approach to empirical work - make sense of the plethora of computational techniques that are now on offer; understand when and why different methods are useful; and see the links that do exist, between them all.