Nested sampling for general Bayesian computation

By John Skilling. 2006

1,247 citations recorded by the source corpus.

Nested sampling estimates directly how the likelihood function relates to prior mass. The evidence (alternatively the marginal likelihood, marginal den- sity of the data, or the prior predictive) is immediately obtained by summation. It is the prime result of the computation, and is accompanied by an estimate of numerical uncertainty. Samples from the posterior distribution are an optional by- product, obtainable for any temperature. The method relies on sampling within a hard constraint on likelihood value, as opposed to the softened likelihood of an- nealing methods. Progress depends only on the shape of the \nested" contours of likelihood, and not on the likelihood values. This invariance (over monotonic re- labelling) allows the method to deal with a class of phase-change problems which eectiv ely defeat thermal annealing.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

A Nested Sampling Algorithm for Cosmological Model Selection (2005, methodology): “Rewriting Eq. (3) in the notation of Skilling (2004) , with X as the fraction of total prior mass such that dX = P(�|H)d� and the likelihood 1 x L(x)”
dynesty: a dynamic nested sampling package for estimating Bayesian posteriors and evidences (2019, background): “Unlike other sampling approaches such as Markov Chain Monte Carlo (MCMC), however, Nested Sampling, contains an additional source of noise arising from our use of noisy estimators X → X̂i of the prior volume at a given iteration i (Skilling 2006).”
Model selection in finite element model updating using the Bayesian evidence statistic (2011, background): “Recently Skilling [30–32] proposed the nested sampling algorithm, which is able to efficiently estimate integrals of the form shown in Eq.”
Applications of Bayesian model selection to cosmological parameters (2005, methodology): “An elegant algorithm called “nested sampling” has been recently put forward by c© 0000 RAS, MNRAS 000, 000–000 Skilling (2004), and implemented in the cosmological context by Bassett et al. (2004); Mukherjee et al. (2006).”
Cobaya: code for Bayesian analysis of hierarchical physical models (2020, background): “PolyChord [3, 4] is a nested sampling [38] algorithm that utilizes slice sampling [39] for sampling within iso-likelihood contours.”
Sandwiching the marginal likelihood using bidirectional Monte Carlo (2015, background): “…ways to compute or approximate partition functions for purposes of model selection (Kass and Raftery, 1995; Meng and Wong, 1996; Gelman and Meng, 1998; Neal, 2001a; Murray and Salakhutdinov, 2008; Attias, 2000; Chib, 1995; Murray and Salakhutdinov, 2009; Skilling, 2006; Wallach et al., 2009).”
Importance Nested Sampling and the MultiNest Algorithm (2013, methodology): “Nested sampling (NS; Skilling 2004, 2006; Sivia and Skilling 2006) is a contemporary Monte Carlo (MC) method targeted at the efﬁcient calculation of the evidence, yet which allows posterior inference as a by-product, providing a means to carry out simultaneous parameter estimation and model…”
Properties of nested sampling (2008, methodology): “Skilling (2006) proposes two approaches: first, a deterministic scheme, where xi is substituted with exp(−i/N) in (2), so that log xi is the expectation of logϕ−1(ϕi); second, a random scheme, where K parallel streams of random numbers xi,k, k = 1, . . . ,K, are generated from the same generating…”
Curvature tension: Evidence for a closed universe (2019, methodology): “[19] John Skilling.”
Testing hypotheses via a mixture estimation model (2014, background): “…of marginal likelihoods is difficult in most settings (Chen et al., 2000; Marin and Robert, 2011) with further controversy about which solution to adopt (Newton and Raftery, 1994; Neal, 1994; Green, 1995; Chib, 1995; Neal, 1999; Skilling, 2006; Steele et al., 2006; Chopin and Robert, 2010).”
Evaluating marginal likelihood with thermodynamic integration method and comparison with several other numerical methods (2016, methodology): “The Xi and Li values are unknown, and they are determined iteratively in the following procedure [Skilling, 2006] based on equation (26): Step 1: Construct an active set of size Nas with parameter samples generated from the prior distribution, calculate for each sample its corresponding joint…”
Inference of the cold dark matter substructure mass function at z = 0.2 using strong gravitational lenses (2014, methodology): “(11) The integral of equation (10) is calculated numerically with the MultiNest method by Feroz & Hobson (2008), which is based on the original nested sampling idea by Skilling (2004).”
Annealing between distributions by averaging moments (2013, background): “While partition function estimation is intractable in general, there has been extensive research on variational [1, 2, 3] and sampling-based [4, 5, 6] approximations.”