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A theoretical and methodological discussion of nested sampling

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Nested sampling is a novel simulation method for approximating marginal likelihoods, proposed by \cite{Skilling:2007a,Skilling:2007b}. We establish that nested sampling leads to an error that vanishes at the standard Monte Carlo rate N-1/2, where N is a tuning parameter that is proportional to the computational effort, and that this error is asymptotically Gaussian. We show that the corresponding asymptotic variance typically grows linearly with the dimension of the parameter. We use these results to discuss the applicability and efficiency of nested sampling in realistic problems, including posterior distributions for mixtures. We propose an extension of nested sampling that makes it possible to avoid resorting to MCMC to obtain the simulated points. We study two alternative methods for computing marginal likelihood, which, in contrast with nested sampling, are based on draws from the posterior distribution and we conduct a comparison with nested sampling on several realistic examples.

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