Constructing summary statistics for approximate Bayesian computation using composite scores

Abstract

For many complex models evaluation of the likelihood is difficult or impossible. Approximate Bayesian computation (ABC) and composite likelihood methods can allow inference to proceed when the likelihood is intractable. If evaluation of the likelihood for some subsets of the data is straightforward, then a composite likelihood can be used as a direct replacement to the likelihood in both Bayesian and frequentist inference. ABC is a likelihood-free method for Bayesian inference, useful in applications where data can easily be simulated from the model. ABC avoids evaluation of the likelihood by comparing summary statistics for the observed data to summary statistics for simulated data. This thesis focuses on developing new methodology for using composite likelihoods within ABC. We propose new methods to construct ABC summary statistics from the sub-scores of a composite likelihood. We provide theoretical support for our approaches by using asymptotic results to relate the summary-based posterior to the corresponding optimally weighted estimating function in frequentist-based inference. We show that the ABC approach avoids calculating the covariances between sub-scores, required in the frequentist framework. Additionally, we show that, by stacking the scores of multiple composite likelihoods, ABC provides a natural method of combining composite likelihoods for applications in which there may be more than one candidate composite likelihood. We illustrate our methods with simulation studies applied to spatial models, time series models, including state-space models, and population genetics models. Our results demonstrate that the new methodology can provide an improved quality of approximate inference, compared with existing Bayesian composite likelihood methods. Our results also illustrate the importance of the choice of sub-scores used to construct the summary statistic. Therefore, an additional contribution is that we propose a new ABC algorithm which aims to adaptively choose a suitable set of sub-scores, in order to obtain the best possible inference while keeping computing costs low.