Search

Sequential Monte Carlo squared (SMC$$^2$$ 2 ) methods can be used for parameter inference of intractable likelihood state-space models. These methods replace the likelihood with an unbiased particle filter estimate, similarly to particle Markov chain Monte Carlo (MCMC). As with particle MCMC, the efficiency of SMC$$^2$$ 2 greatly depends on the variance of the likelihood estimator, and therefore on the number of state particles used within the particle filter. We introduce novel methods to adaptively select the number of state particles within SMC$$^2$$ 2 using the expected squared jumping distance to trigger the adaptation, and modifying the exchange importance sampling method of Chopin et al. (J R Stat Soc: Ser B (Stat Method) 75(3):397–426, 2012) to replace the current set of state particles with the new set of state particles. The resulting algorithm is fully automatic, and can significantly improve current methods. Code for our methods is available at https://github.com/imkebotha/adaptive-exact-approximate-smc.

We review Skilling's nested sampling (NS) algorithm for Bayesian inference and more broadly multi-dimensional integration. After recapitulating the principles of NS, we survey developments in implementing efficient NS algorithms in practice in high-dimensions, including methods for sampling from the so-called constrained prior. We outline the ways in which NS may be applied and describe the application of NS in three scientific fields in which the algorithm has proved to be useful: cosmology, gravitational-wave astronomy, and materials science. We close by making recommendations for best practice when using NS and by summarizing potential limitations and optimizations of NS., Comment: 20 pages + supplementary information, 5 figures. preprint version; published version at https://www.nature.com/articles/s43586-022-00121-x