Nested sampling algorithm

• Bayesian inference
• Bayesian probability
• Bayes' theorem
• Bernstein–von Mises theorem
• Coherence
• Cox's theorem
• Cromwell's rule
• Likelihood principle
• Principle of indifference
• Principle of maximum entropy

• Conjugate prior
• Linear regression
• Empirical Bayes
• Hierarchical model

• Markov chain Monte Carlo
• Laplace's approximation
• Integrated nested Laplace approximations
• Variational inference
• Approximate Bayesian computation

• Bayes estimator
• Credible interval
• Maximum a posteriori estimation

• Evidence lower bound
• Nested sampling

• Bayes factor (Schwarz criterion)
• Model averaging
• Posterior predictive

• Mathematicsportal

The posterior probability for M1 and M1 has a set of parameters that can be grouped together and called θ. The marginalization for M2. This integral is often analytically intractable, and in these cases it is necessary to employ a numerical algorithm to find an approximation. The nested sampling algorithm was developed by John Skilling specifically to approximate these marginalization integrals, and it has the added benefit of generating samples from the posterior distribution M is N points j do % The number of iterations j is chosen by guesswork. Xi:=exp⁡(−i/N); wi. Update the point with least likelihood by sampling from the prior restricted to likelihoods above Xi is an estimate of the amount of prior mass covered by the hypervolume in parameter space of all points with likelihood greater than {θ∣P(D∣θ,M)=P(D∣θi−1,M)} and i of j→∞, this estimator has a positive bias of order exp⁡(−1/N) in the above algorithm.

The idea is to subdivide the range of θ would map to this interval. This can be thought of as a Bayesian's way to numerically implement Lebesgue integration.

The point with least likelihood can be updated with some Markov chain Monte Carlo steps according to the prior, accepting only steps that keep the likelihood above

1. ↑ Skilling, John (2004). "Nested Sampling". AIP Conference Proceedings. 735: 395–405. Bibcode:2004AIPC..735..395S. doi:10.1063/1.1835238.
2. ↑ Walter, Clement (2017). "Point-process based Monte Carlo estimation". Statistics and Computing. 27: 219–236. arXiv:1412.6368. doi:10.1007/s11222-015-9617-y. S2CID 14639080.
3. 1 2 Buchner, Johannes (1 January 2023). "Nested sampling methods". Statistics Surveys. 17: 169. arXiv:2101.09675. Bibcode:2023StSur..17..169B. doi:10.1214/23-SS144.
4. ↑ Mukherjee, P.; Parkinson, D.; Liddle, A.R. (2006). "A Nested Sampling Algorithm for Cosmological Model Selection". Astrophysical Journal. 638 (2): 51–54. arXiv:astro-ph/0508461. Bibcode:2006ApJ...638L..51M. doi:10.1086/501068. S2CID 6208051.
5. 1 2 Feroz, F.; Hobson, M.P. (2008). "Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses". MNRAS. 384 (2): 449–463. arXiv:0704.3704. Bibcode:2008MNRAS.384..449F. doi:10.1111/j.1365-2966.2007.12353.x. S2CID 14226032.
6. ↑ Mthembu, L.; Marwala, T.; Friswell, M.I.; Adhikari, S. (2011). "Model selection in finite element model updating using the Bayesian evidence statistic". Mechanical Systems and Signal Processing. 25 (7): 2399–2412. Bibcode:2011MSSP...25.2399M. doi:10.1016/j.ymssp.2011.04.001.
7. ↑ Partay, Livia B. (2010). "Efficient Sampling of Atomic Configurational Spaces". The Journal of Physical Chemistry B. 114 (32): 10502–10512. arXiv:0906.3544. Bibcode:2010JPCB..11410502P. doi:10.1021/jp1012973. PMID 20701382. S2CID 16834142.
8. ↑ Buchner, Johannes (February 2024). "Relative Jump Distance: A diagnostic for Nested Sampling". arXiv:2402.11936 [stat.ME].
9. ↑ Talts, Sean; Betancourt, Michael; Simpson, Daniel; Vehtari, Aki; Gelman, Andrew (April 2018). "Validating Bayesian Inference Algorithms with Simulation-Based Calibration". arXiv:1804.06788 [stat.ME].
10. ↑ Higson, Edward (2018). "dyPolyChord: dynamic nested sampling with PolyChord". Journal of Open Source Software. 3 (29): 965. doi:10.21105/joss.00965.
11. ↑ Günther, Maximilian; et al. (2019). "A super-Earth and two sub-Neptunes transiting the nearby and quiet M dwarf TOI-270". Nature Astronomy. 3 (12): 1099–1108. arXiv:1903.06107. Bibcode:2019NatAs...3.1099G. doi:10.1038/s41550-019-0845-5. S2CID 119286334.