ABSTRACT
In Bayesian inference, the posterior distribution for parameters θ ∈ Θ is given by π(θ | y) ∝ π(y | θ)π(θ), where one’s prior beliefs about the unknownparameters, as expressed through the prior distribution π(θ), are updated by the observed data y ∈ Y via the likelihood function π(y | θ). Inference for the parameters θ is then based on the posterior distribution. Except in simple cases, numerical simulation methods, such as Markov chain Monte Carlo (MCMC), are required to approximate the integrations needed to summarize features of the posterior distribution. Inevitably, increasing demands on statistical modeling and computation have resulted in the development of progressively more sophisticated algorithms. Most recently there has been interest in performing Bayesian analyses for models which
are sufficiently complex that the likelihood function π(y | θ) is either analytically unavailable or computationally prohibitive to evaluate. The classes of algorithms and methods developed to perform Bayesian inference in this setting have become known as likelihoodfree computation or approximate Bayesian computation (Beaumont et al., 2002; Marjoram et al., 2003; Ratmann et al., 2009; Sisson et al., 2007; Tavaré et al., 1997). This name refers to the circumventingof explicit evaluationof the likelihoodbya simulation-based approximation. Likelihood-free methods are rapidly gaining popularity as a practical approach to fitting
models under the Bayesian paradigm that would otherwise have been computationally impractical. To date they have found widespread usage in a diverse range of applications. These include wireless communications engineering (Nevat et al., 2008), quantile distributions (Drovandi and Pettitt, 2009), HIV contact tracing (Blum and Tran, 2010), the evolution of drug resistance in tuberculosis (Luciani et al., 2009), population genetics (Beaumont et al., 2002), protein networks (Ratmann et al., 2009, 2007), archeology (Wilkinson and Tavaré, 2009); ecology (Jabot and Chave, 2009), operational risk (Peters and Sisson, 2006), species migration (Hamilton et al., 2005), chain-ladder claims reserving (Peters et al., 2008), coalescent models (Tavaré et al., 1997), α-stable models (Peters et al., in press), models for extremes (Bortot et al., 2007), susceptible-infected-removed models (Toni et al., 2009), pathogen transmission (Tanaka et al., 2006), and human evolution (Fagundes et al., 2007).
ALGORITHM 12.1 LIKELIHOOD-FREE REJECTION SAMPLING ALGORITHM
