Constructing summary statistics from composite likelihoods for approximate Bayesian computation

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To generate realisations from a posterior distribution, methods such as Markov chain Monte Carlo and sequential Monte Carlo rely on evaluation of the likelihood function. However, for many complex models the likelihood is computationally intractable. Approximate Bayesian computation (ABC) and composite likelihoods can be useful tools to allow inference to proceed when the likelihood is not available. When data can easily be simulated from a model, ABC provides an approximation to the posterior distribution, avoiding evaluation of the likelihood, by measuring the similarity between the observed data and simulated data. A basic ABC rejection scheme simulates data using parameter values sampled from the prior distribution, and accepts the parameter values if the distance between vectors of summary statistics for the observed data and simulated data is smaller than some pre-defined tolerance. The choice of summary statistics used within ABC can greatly affect the quality of the approximate inference. If the full likelihood is unavailable, but evaluation of the likelihood for some subsets of the data is straightforward, then a composite likelihood can be used to replace the full likelihood. A composite likelihood is a weighted product of valid likelihood terms, corresponding to a collection of marginal or conditional events. The weights in a composite likelihood can be set equal and ignored. However, a carefully selected set of weights can improve statistical efficiency. If a composite likelihood is used directly in Bayes' theorem the variability in the posterior is often greatly underestimated, with the composite posterior being excessively concentrated. Calibration methods have been proposed to adjust the composite likelihood for use in Bayes' theorem. However, such methods can result in the calibrated composite posterior being too dispersed. Previous work combining composite likelihoods and ABC suggests using the composite score as the summary statistic for ABC. To implement this approach a suitable composite likelihood, including the associated weights, must first be chosen. For some models there may not be a clear choice for the weights, or there may be several candidate composite likelihoods to choose from. The aims of this project are: to explore new methods for constructing summary statistics for ABC from composite likelihoods, to develop methods for constructing summary statistics such that the associated weights in the composite likelihood can be chosen automatically, to develop techniques that combine multiple composite likelihoods for contribution to summary statistics, and to ultimately develop a thorough methodology that will allow automatic construction of a summary statistic to use within ABC to allow efficient and accurate inference for models for which at least one composite likelihood can be defined. Possible applications for this new methodology include spatial extremes and space-time models.