Unbiased and Consistent Nested Sampling via Sequential Monte Carlo

Abstract

We introduce a new class of sequential Monte Carlo methods which reformulates the essence of the nested sampling method of Skilling (2006) in terms of sequential Monte Carlo techniques. Two new algorithms are proposed, nested sampling via sequential Monte Carlo (NS-SMC) and adaptive nested sampling via sequential Monte Carlo (ANS-SMC). The new framework allows convergence results to be obtained in the setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An additional benefit is that marginal likelihood (normalising constant) estimates given by NS-SMC are unbiased. In contrast to NS, the analysis of our proposed algorithms does not require the (unrealistic) assumption that the simulated samples be independent. We show that a minor adjustment to our ANS-SMC algorithm recovers the original NS algorithm, which provides insights as to why NS seems to produce accurate estimates despite a typical violation of its assumptions. A numerical study is conducted where the performance of the proposed algorithms and temperature-annealed SMC is compared on challenging problems. Code for the experiments is made available online at https://github.com/LeahPrice/SMC-NS .