A pseudo-marginal sequential Monte Carlo algorithm for random effects models in Bayesian sequential design
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Motivated by the need to sequentially design experiments for the collection of data in batches or blocks, a new pseudo-marginal sequential Monte Carlo algorithm is proposed for random effects models where the likelihood is not analytic, and has to be approximated. This new algorithm is an extension of the idealised sequential Monte Carlo algorithm where we propose to unbiasedly approximate the likelihood to yield an efficient exact-approximate algorithm to perform inference and make decisions within Bayesian sequential design. We propose four approaches to unbiasedly approximate the likelihood: standard Monte Carlo integration; randomised quasi-Monte Carlo integration, Laplace importance sampling and a combination of Laplace importance sampling and randomised quasi-Monte Carlo. These four methods are compared in terms of the estimates of likelihood weights and in the selection of the optimal sequential designs in an important pharmacological study related to the treatment of critically ill patients. As the approaches considered to approximate the likelihood can be computationally expensive, we exploit parallel computational architectures to ensure designs are derived in a timely manner.
KeywordsGraphics processing unit Importance Sampling Intractable likelihood Laplace approximation Nonlinear regression Optimal design Parallel computing Particle filter Randomised quasi Monte Carlo
This work was supported by the Australian Research Council Centre of Excellence for Mathematical & Statistical Frontiers. The work of A.N. Pettitt was supported by an ARC Discovery Project (DP110100159), and the work of J.M. McGree was supported by an ARC Discovery Project (DP120100269). We would also like to thank the two referees who offered helpful comments to improve the article.
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