Nested sampling for general bayesian computation

Bayesian model selection provides the cosmologist with an exacting tool to distinguish between competing models based purely on the data via the Bayesian evidence. Previous methods to calculate this quantity either lacked general applicability or were computationally demanding. However, nested sampling, which was recently applied successfully to cosmology by Muhkerjee et al., overcomes both of these impediments. Their implementation restricts the parameter space sampled, and thus improves the efficiency, using a shrinking ellipsoidal bound in the n-dimensional parameter space encompassing parameter samples above an increasing likelihood threshold. However, if the likelihood function contains any multimodality, separated over a significant portion of the parameter space, then the ellipse is prevented from constraining the sampling region efficiently. In this paper, we introduce a method of clustered nested sampling whereby multiple ellipsoidal clusters can form over any region of the prior volume - thus improving the efficiency by a factor which is equal to the ratio of the volumes enclosed by the set of small clustered ellipsoids and the large single ellipse that would necessarily be required without clustering. In addition, we have implemented a method for determining the expectation and variance of the final evidence value without the need to use sampling error from repetitions of the algorithm; this further reduces the computational load by at least an order of magnitude. It should be noted that this latter development is relevant for all cosmological applications, not just for those involving multimodal posteriors. We have applied our algorithm to a pair of toy models and one cosmological example where we demonstrate that the number of likelihood evaluations required is similar to 4 per cent of that necessary using previous algorithms. We have produced a FORTAN library containing our routines which can be called from any sampling code, in addition for convenience we have incorporated it into the popular COSMOMC code as COSMOCLUST. Both are available for download at http://www.mrao.cam.ac.uk/software/cosmoclust.