Bayesian Methods for Astrophysics and Particle Physics

Chapter 2
Multimodal Nested Sampling
The content of this chapter is also presented in Feroz & Hobson (2008).
2.1 Abstract
In performing a Bayesian analysis of astronomical data, two difficult problems
often emerge. First, in estimating the parameters of some model for the data,
the resulting posterior distribution may be multimodal or exhibit pronounced
(curving) degeneracies, which can cause problems for traditional Markov Chain
Monte Carlo (MCMC) sampling methods. Second, in selecting between a set of
competing models, calculation of the Bayesian evidence for each model is compu-
tationally expensive using existing methods such as thermodynamic integration.
The nested sampling method introduced by Skilling (2004), has greatly reduced
the computational expense of calculating evidences and also produces posterior
inferences as a by-product. This method has been applied successfully in cos-
mological applications by Mukherjee et al. (2006), but their implementation was
efficient only for unimodal distributions without pronounced degeneracies. Shaw
et al. (2006b) recently introduced a clustered nested sampling method which is
significantly more efficient in sampling from multimodal posteriors and also de-
termines the expectation and variance of the final evidence from a single run of
the algorithm, hence providing a further increase in efficiency. In this work, we
build on the work of Shaw et al. (2006b) and present three new methods for
sampling and evidence evaluation from distributions that may contain multiple
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