- Title
- New prior sampling methods and equidistribution testing for nested sampling
- Creator
- Stokes, Barrie James
- Relation
- University of Newcastle Research Higher Degree Thesis
- Resource Type
- thesis
- Date
- 2018
- Description
- Research Doctorate - Doctor of Philosophy (PhD)
- Description
- Nested Sampling is a Monte Carlo-based algorithm for fitting statistical models to data in the Bayesian setting, introduced by John Skilling. Its key feature is direct estimation of the Evidence Ζ and the Information H associated with a particular model, prior distribution and dataset. David MacKay has said that “The evidence Ζ is often the single most important number in the [Bayesian] problem and I think every effort should be devoted to calculating it.” John Skilling claims that ”Nested sampling does this by giving a direct estimate of the density of states. Posterior samples are an optional by-product.” The central problem of Nested Sampling—likelihood-restricted prior sampling—is the drawing of random samples from the prior distribu- tion over nested likelihood-restricted regions in the problem parameter space. This work is addressed to (i) the development of new algorithms for solving this central problem, and (ii) devising and validating new statistical tests of equidistribution of sample point datasets, verifying that new likelihood-restricted prior samplers generate correctly (usually uniformly) distributed random samples. Two new prior samplers have been proposed and tested, one based on kernel mixture distributions, and the other on the Nelder-Mead optimisation algorithm. Two equidistribution tests based on nearest neighbours distances (NNDs) of uniform points in Euclidean space have been developed. The first is the Bayes Factor Equidistribution (BFEQD) test, and the second combines a newly devised orthodox test of the equidistribu- tion of a pair of scalar datasets—the Area of Interest Randomisation Equidistribution (AoIRE) test—with a previously published Bayesian test of the uniformity of samples in the interval (0,1)—the Sivia-Rawlings Bayesian Uniformity (SRBU) test—to produce the hybrid AoIRE- SRBU test of a likelihood-restricted prior sampling algorithm. Sampling properties of the BFEQD, AoIRE, and SRBU tests have been investigated, and power testing under the Null hypothesis H₀ carried out. Additional contributions to the field are (i) a Computer Algebra System (CAS) validation of John Skilling’s proof of Nested Sampling’s convergence, (ii) a novel CAS-based derivation of the general PDF of the NNDs of order d of uniformly distributed points in n dimensions with spatial density λ, and (iii) an investigation into, and prior sampling simulation testing of, a random sampling algorithm—Hit-and-Run— first published by Turcin in 1971.
- Subject
- nested sampling; Bayesian model fitting; likelihood-restricted prior sampling; equidistribution testing
- Identifier
- http://hdl.handle.net/1959.13/1391811
- Identifier
- uon:33299
- Rights
- Copyright 2018 Barrie James Stokes
- Language
- eng
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| View Details Download | ATTACHMENT03 | Abstract | 109 KB | Adobe Acrobat PDF | View Details Download | ||
| View Details Download | ATTACHMENT04 | Read me | 68 KB | Adobe Acrobat PDF | View Details Download |