In performing a Bayesian analysis of astronomical data, two difficultproblems often emerge. First, in estimating the parameters of some model forthe data, the resulting posterior distribution may be multimodal or exhibitpronounced (curving) degeneracies, which can cause problems for traditionalMCMC sampling methods. Second, in selecting between a set of competing models,calculation of the Bayesian evidence for each model is computationallyexpensive. The nested sampling method introduced by Skilling (2004), hasgreatly reduced the computational expense of calculating evidences and alsoproduces posterior inferences as a by-product. This method has been appliedsuccessfully in cosmological applications by Mukherjee et al. (2006), but theirimplementation was efficient only for unimodal distributions without pronounceddegeneracies. Shaw et al. (2007), recently introduced a clustered nestedsampling method which is significantly more efficient in sampling frommultimodal posteriors and also determines the expectation and variance of thefinal evidence from a single run of the algorithm, hence providing a furtherincrease in efficiency. In this paper, we build on the work of Shaw et al. andpresent three new methods for sampling and evidence evaluation fromdistributions that may contain multiple modes and significant degeneracies; wealso present an even more efficient technique for estimating the uncertainty onthe evaluated evidence. These methods lead to a further substantial improvementin sampling efficiency and robustness, and are applied to toy problems todemonstrate the accuracy and economy of the evidence calculation and parameterestimation. Finally, we discuss the use of these methods in performing Bayesianobject detection in astronomical datasets.
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