Unbiased and Consistent Nested Sampling via Sequential Monte Carlo (1805.03924v8)
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 .
- Alexandroff P (1924) Théorie des ensembles. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 178:185–187
- Botev ZI, Kroese DP (2012) Efficient Monte Carlo simulation via the generalized splitting method. Statistics and Computing 22(1):1–16
- Brewer BJ (2014) Inference for trans-dimensional Bayesian models with diffusive nested sampling. arXiv:14113921
- Cérou F, Guyader A (2016) Fluctuation analysis of adaptive multilevel splitting. Annals of Applied Probability 26(6):3319–3380
- Chopin N, Papaspiliopoulos O (2020) An introduction to sequential Monte Carlo. Springer
- Chopin N, Robert CP (2010) Properties of nested sampling. Biometrika 97(3):741–755
- Dau H, Chopin N (2022) Waste-free sequential Monte Carlo. Journal of the Royal Statistical Society Series B 84(1):114–148
- Doucet A, Johansen AM (2011) A Tutorial on Particle filtering and smoothing: Fifteen years later. The Oxford handbook of nonlinear filtering pp 656–705
- Dudley R (2002) Real Analysis and Probability. Cambridge University Press
- Evans M (2007) Discussion of nested sampling for Bayesian computations by John Skilling. Bayesian Statistics 8:491–524
- Kahn H, Harris TE (1951) Estimation of particle Transmission by Random Sampling. National Bureau of Standards Applied Mathematics Series 12:27–30
- Lopes HF, West M (2004) Bayesian model assessment in factor analysis. Statistica Sinica 14(1):41–67
- Neal RM (2001) Annealed importance sampling. Statistics and Computing 11(2):125–139
- Pasarica C, Gelman A (2010) Adaptively scaling the Metropolis Hastings algorithm using expected squared jumped distance. Statistica Sinica 20(1):343–364
- Rudin W (1964) Principles of mathematical analysis, vol 3. McGraw-hill New York
- Serfozo R (1982) Convergence of Lebesgue integrals with varying measures. Sankhyā: The Indian Journal of Statistics, Series A 44(3):380–402
- Skilling J (2006) Nested sampling for general Bayesian computation. Bayesian Analysis 1(4):833–859
- Vegetti S, Koopmans LVE (2009) Bayesian strong gravitational-lens modelling on adaptive grids: objective detection of mass substructure in galaxies. Monthly Notices of the Royal Astronomical Society 392(3):945–963
- Villani A (1985) Another note on the inclusion Lp(μ)⊂Lq(μ)superscript𝐿𝑝𝜇superscript𝐿𝑞𝜇{L}^{p}(\mu)\subset L^{q}(\mu)italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_μ ) ⊂ italic_L start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_μ ). The American Mathematical Monthly 92(7):485–C76
- Walter C (2017) Point process-based Monte Carlo estimation. Statistics and Computing 27(1):219–236
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