Multimodal nested sampling: an efficient and robust alternative to MCMC methods for astronomical data analysis

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 MCMC sampling methods. Second, in selecting between a set of competing models, calculation of the Bayesian evidence for each model is computationally expensive. 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 cosmological applications by Mukherjee et al. (2006), but their implementation was efficient only for unimodal distributions without pronounced degeneracies. Shaw et al. (2007), recently introduced a clustered nested sampling method which is significantly more efficient in sampling from multimodal posteriors and also determines the expectation and variance of the final evidence from a single run of the algorithm, hence providing a further increase in efficiency. In this paper, we build on the work of Shaw et al. and present three new methods for sampling and evidence evaluation from distributions that may contain multiple modes and significant degeneracies; we also present an even more efficient technique for estimating the uncertainty on the evaluated evidence. These methods lead to a further substantial improvement in sampling efficiency and robustness, and are applied to toy problems to demonstrate the accuracy and economy of the evidence calculation and parameter estimation. Finally, we discuss the use of these methods in performing Bayesian object detection in astronomical datasets.Comment: 14 pages, 11 figures, submitted to MNRAS, some major additions to the previous version in response to the referee's comment

Linking the genomic signatures of human beat synchronization and learned song in birds

The development of rhythmicity is foundational to communicative and social behaviours in humans and many other species, and mechanisms of synchrony could be conserved across species. The goal of the current paper is to explore evolutionary hypotheses linking vocal learning and beat synchronization through genomic approaches, testing the prediction that genetic underpinnings of birdsong also contribute to the aetiology of human interactions with musical beat structure. We combined state-of-the-art-genomic datasets that account for underlying polygenicity of these traits: birdsong genome-wide transcriptomics linked to singing in zebra finches, and a human genome-wide association study of beat synchronization. Results of competitive gene set analysis revealed that the genetic architecture of human beat synchronization is significantly enriched for birdsong genes expressed in songbird Area X (a key nucleus for vocal learning, and homologous to human basal ganglia). These findings complement ethological and neural evidence of the relationship between vocal learning and beat synchronization, supporting a framework of some degree of common genomic substrates underlying rhythm-related behaviours in two clades, humans and songbirds (the largest evolutionary radiation of vocal learners). Future cross-species approaches investigating the genetic underpinnings of beat synchronization in a broad evolutionary context are discussed

Efficient Bayesian inference for multimodal problems in cosmology

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, then the ellipse is prevented from constraining the sampling region efficiently. In this paper we introduce a method of clustered ellipsoidal nested sampling which can form multiple ellipses around each individual peak in the likelihood. 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. 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 ~ 4% of that necessary for using previous algorithms. We have produced a fortran 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 .Comment: 7 pages, 8 figures, changed to match version accepted by MNRA

The spectral action and cosmic topology

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincare' dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and see we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincare' homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces.Comment: 55 pages, LaTe

Genome-wide association study of musical beat synchronization demonstrates high polygenicity

Moving in synchrony to the beat is a fundamental component of musicality. Here we conducted a genome-wide association study to identify common genetic variants associated with beat synchronization in 606,825 individuals. Beat synchronization exhibited a highly polygenic architecture, with 69 loci reaching genome-wide significance (P < 5 × 10−8) and single-nucleotide-polymorphism-based heritability (on the liability scale) of 13%–16%. Heritability was enriched for genes expressed in brain tissues and for fetal and adult brain-specific gene regulatory elements, underscoring the role of central-nervous-system-expressed genes linked to the genetic basis of the trait. We performed validations of the self-report phenotype (through separate experiments) and of the genome-wide association study (polygenic scores for beat synchronization were associated with patients algorithmically classified as musicians in medical records of a separate biobank). Genetic correlations with breathing function, motor function, processing speed and chronotype suggest shared genetic architecture with beat synchronization and provide avenues for new phenotypic and genetic explorations

Localization phenomena in Nonlinear Schrodinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates

We study the properties of the ground state of Nonlinear Schr\"odinger Equations with spatially inhomogeneous interactions and show that it experiences a strong localization on the spatial region where the interactions vanish. At the same time, tunneling to regions with positive values of the interactions is strongly supressed by the nonlinear interactions and as the number of particles is increased it saturates in the region of finite interaction values. The chemical potential has a cutoff value in these systems and thus takes values on a finite interval. The applicability of the phenomenon to Bose-Einstein condensates is discussed in detail

The current status of observational cosmology

Observational cosmology has indeed made very rapid progress in recent years. The ability to quantify the universe has largely improved due to observational constraints coming from structure formation. The transition to precision cosmology has been spearheaded by measurements of the anisotropy in the cosmic microwave background (CMB) over the past decade. Observations of the large scale structure in the distribution of galaxies, high red-shift supernova, have provided the required complementary information. We review the current status of cosmological parameter estimates from joint analysis of CMB anisotropy and large scale structure (LSS) data. We also sound a note of caution on overstating the successes achieved thus far.Comment: 13 pages, 3 figures, Latex style files included, To appear in the proceedings of ICGC-04. Minor rewording in the abstract and introductio

Solitary waves in coupled nonlinear Schrodinger equations with spatially inhomogeneous nonlinearities

Using Lie group theory we construct explicit solitary wave solutions of coupled nonlinear Schrodinger systems with spatially inhomogeneous nonlinearities. We present the general theory, use it to construct different families of explicit solutions and study their linear and dynamical stability

Cosmic Topology of Polyhedral Double-Action Manifolds

A special class of non-trivial topologies of the spherical space S^3 is investigated with respect to their cosmic microwave background (CMB) anisotropies. The observed correlations of the anisotropies on the CMB sky possess on large separation angles surprising low amplitudes which might be naturally be explained by models of the Universe having a multiconnected spatial space. We analysed in CQG 29(2012)215005 the CMB properties of prism double-action manifolds that are generated by a binary dihedral group D^*_p and a cyclic group Z_n up to a group order of 180. Here we extend the CMB analysis to polyhedral double-action manifolds which are generated by the three binary polyhedral groups (T^*, O^*, I^*) and a cyclic group Z_n up to a group order of 1000. There are 20 such polyhedral double-action manifolds. Some of them turn out to have even lower CMB correlations on large angles than the Poincare dodecahedron