[Paper Review] Clustered nested sampling: efficient Bayesian inference for cosmology

Ruth G. Shaw, M. Bridges+1 more|arXiv (Cornell University)|Jan 30, 2007

Bayesian Methods and Mixture Models6 citations

TL;DR

This paper introduces clustered nested sampling, a method that enhances Bayesian evidence computation in cosmology by replacing a single large ellipsoidal constraint with multiple smaller ellipsoids centered on distinct likelihood peaks. By adapting to multi-modal likelihoods, the approach improves sampling efficiency by a factor equal to the ratio of the total volume of small clusters to the volume of a single enclosing ellipse.

ABSTRACT

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 (Skilling 2004), which was recently applied successfully to cosmology by Muhkerjee et al. 2006, overcomes both of these impediments. Their implementation restricts the parameter space sampled, and thus improves the efficiency, using a decreasing ellipsoidal bound in the n-dimensional parameter space centred on the maximum likelihood point. However, if the likelihood function contains any multi-modality, separated over a significant portion of the parameter space then the ellipse is prevented from constraining the sampling region by less than the distance between the likelihood peaks. In this paper we introduce a method of clustered nested sampling whereby ellipsoidal clusters can form on any peaks identified –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

Motivation & Objective

• Address the inefficiency of standard nested sampling in multi-modal parameter spaces common in cosmological model selection.
• Overcome the limitation of single-ellipsoid constraints that fail to explore separated likelihood peaks.
• Develop a method that dynamically identifies and samples around multiple likelihood peaks to improve computational efficiency.
• Enable more accurate and scalable Bayesian evidence evaluation for complex cosmological models.

Proposed method

• Replace the standard single ellipsoidal constraint in nested sampling with multiple smaller ellipsoids centered on identified likelihood peaks.
• Use a clustering algorithm to detect and group parameter space regions corresponding to distinct local maxima in the likelihood function.
• Apply nested sampling independently within each cluster to explore high-likelihood regions more effectively.
• Dynamically update cluster boundaries based on the distribution of live points and likelihood gradients.
• Combine evidence contributions from all clusters to compute the total Bayesian evidence.
• Integrate a method for determining cluster boundaries and peak identification without prior knowledge of the likelihood structure.

Experimental results

Research questions

• RQ1 How can nested sampling be made more efficient in the presence of multi-modal likelihood functions in cosmological parameter inference?
• RQ2 What is the computational advantage of using multiple ellipsoidal clusters over a single enclosing ellipsoid in nested sampling?
• RQ3 Can clustering improve the accuracy and convergence of Bayesian evidence estimation in models with separated likelihood peaks?
• RQ4 How can likelihood peaks be reliably detected and isolated in high-dimensional parameter spaces without prior knowledge?

Key findings

• Clustered nested sampling improves sampling efficiency by a factor equal to the ratio of the total volume of small clusters to the volume of a single enclosing ellipse.
• The method effectively samples multi-modal likelihoods where standard nested sampling fails due to trapped ellipsoidal constraints.
• The approach enables more accurate Bayesian evidence computation in cosmological models with complex, separated likelihood structures.
• The implementation includes a mechanism for identifying likelihood peaks and forming clusters without requiring prior knowledge of the parameter space topology.