Search

We review Skilling's nested sampling (NS) algorithm for Bayesian inference and more broadly multi-dimensional integration. After recapitulating the principles of NS, we survey developments in implementing efficient NS algorithms in practice in high-dimensions, including methods for sampling from the so-called constrained prior. We outline the ways in which NS may be applied and describe the application of NS in three scientific fields in which the algorithm has proved to be useful: cosmology, gravitational-wave astronomy, and materials science. We close by making recommendations for best practice when using NS and by summarizing potential limitations and optimizations of NS., Comment: 20 pages + supplementary information, 5 figures. preprint version; published version at https://www.nature.com/articles/s43586-022-00121-x

Priors in Bayesian analyses often encode informative domain knowledge that can be useful in making the inference process more efficient. Occasionally, however, priors may be unrepresentative of the parameter values for a given dataset, which can result in inefficient parameter space exploration, or even incorrect inferences, particularly for nested sampling (NS) algorithms. Simply broadening the prior in such cases may be inappropriate or impossible in some applications. Hence our previous solution to this problem, known as posterior repartitioning (PR), redefines the prior and likelihood while keeping their product fixed, so that the posterior inferences and evidence estimates remain unchanged, but the efficiency of the NS process is significantly increased. In its most practical form, PR raises the prior to some power beta, which is introduced as an auxiliary variable that must be determined on a case-by-case basis, usually by lowering beta from unity according to some pre-defined `annealing schedule' until the resulting inferences converge to a consistent solution. Here we present a very simple yet powerful alternative Bayesian approach, in which beta is instead treated as a hyperparameter that is inferred from the data alongside the original parameters of the problem, and then marginalised over to obtain the final inference. We show through numerical examples that this Bayesian PR (BPR) method provides a very robust, self-adapting and computationally efficient `hands-off' solution to the problem of unrepresentative priors in Bayesian inference using NS. Moreover, unlike the original PR method, we show that even for representative priors BPR has a negligible computational overhead relative to standard nesting sampling, which suggests that it should be used as the default in all NS analyses.

The Arcminute Microkelvin Imager (AMI) carried out a blind survey for galaxy clusters via their Sunyaev-Zel'dovich effect decrements between 2008 and 2011. The first detection, known as AMI-CL J0300+2613, has been reobserved with AMI equipped with a new digital correlator with high dynamic range. The combination of the new AMI data and more recent high-resolution sub-mm and infra-red maps now shows the feature in fact to be a ring of positive dust-correlated Galactic emission, which is likely to be anomalous microwave emission (AME). If so, this is the first completely blind detection of AME at arcminute scales., Comment: 11 pages, 13 figures, 4 tables. Submitted to MNRAS

A new method is presented for modelling the physical properties of galaxy clusters. Our technique moves away from the traditional approach of assuming specific parameterised functional forms for the variation of physical quantities within the cluster, and instead allows for a 'free-form' reconstruction, but one for which the level of complexity is determined automatically by the observational data and may depend on position within the cluster. This is achieved by representing each independent cluster property as some interpolating or approximating function that is specified by a set of control points, or 'nodes', for which the number of nodes, together with their positions and amplitudes, are allowed to vary and are inferred in a Bayesian manner from the data. We illustrate our nodal approach in the case of a spherical cluster by modelling the electron pressure profile Pe(r) in analyses both of simulated Sunyaev-Zel'dovich (SZ) data from the Arcminute MicroKelvin Imager (AMI) and of real AMI observations of the cluster MACS J0744+3927 in the CLASH sample. We demonstrate that one may indeed determine the complexity supported by the data in the reconstructed Pe(r), and that one may constrain two very important quantities in such an analysis: the cluster total volume integrated Comptonisation parameter (Ytot) and the extent of the gas distribution in the cluster (rmax). The approach is also well-suited to detecting clusters in blind SZ surveys., Comment: 11 pages, 11 figures, 1 table

Using Arcminute Microkelvin Imager (AMI) SZ observations towards ten CLASH clusters we investigate the influence of cluster mergers on observational galaxy cluster studies. Although selected to be largely relaxed, there is disagreement in the literature on the dynamical states of CLASH sample members. We analyse our AMI data in a fully Bayesian way to produce estimated cluster parameters and consider the intrinsic correlations in our NFW/GNFW-based model. Varying pressure profile shape parameters, illustrating an influence of mergers on scaling relations, induces small deviations from the canonical self-similar predictions -- in agreement with simulations of Poole et al. 2007 who found that merger activity causes only small scatter perpendicular to the relations. We demonstrate this effect observationally using the different dependencies of SZ and X-ray signals to $n_{\rm e}$ that cause different sensitivities to the shocking and/or fractionation produced by mergers. Plotting $Y_{\rm X}$--$M_{\rm gas}$ relations (where $Y_{\rm X}=M_{\rm gas}T$) derived from AMI SZ and from $Chandra$ X-ray gives ratios of AMI and $Chandra$ $Y_{\rm X}$ and $M_{\rm gas}$ estimates that indicate movement of clusters \textit{along} the scaling relation, as predicted by Poole et al. 2007. Clusters that have moved most along the relation have the most discrepant $T_{\rm SZ}$ and $T_{\rm X}$ estimates: all the other clusters (apart from one) have SZ and X-ray estimates of $M_{\rm gas}$, $T$ and $Y_{\rm X}$ that agree within $r_{500}$. We use SZ vs X-ray discrepancies in conjunction with $Chandra$ maps and $T_{\rm X}$ profiles, making comparisons with simulated cluster merger maps in Poole et al. 2006, to identify disturbed members of our sample and estimate merger stages., Comment: 22 pages, 15 figures, 7 tables

The Advanced LIGO and Advanced Virgo gravitational wave (GW) detectors will begin operation in the coming years, with compact binary coalescence events a likely source for the first detections. The gravitational waveforms emitted directly encode information about the sources, including the masses and spins of the compact objects. Recovering the physical parameters of the sources from the GW observations is a key analysis task. This work describes the LALInference software library for Bayesian parameter estimation of compact binary signals, which builds on several previous methods to provide a well-tested toolkit which has already been used for several studies. We show that our implementation is able to correctly recover the parameters of compact binary signals from simulated data from the advanced GW detectors. We demonstrate this with a detailed comparison on three compact binary systems: a binary neutron star, a neutron star black hole binary and a binary black hole, where we show a cross-comparison of results obtained using three independent sampling algorithms. These systems were analysed with non-spinning, aligned spin and generic spin configurations respectively, showing that consistent results can be obtained even with the full 15-dimensional parameter space of the generic spin configurations. We also demonstrate statistically that the Bayesian credible intervals we recover correspond to frequentist confidence intervals under correct prior assumptions by analysing a set of 100 signals drawn from the prior. We discuss the computational cost of these algorithms, and describe the general and problem-specific sampling techniques we have used to improve the efficiency of sampling the compact binary coalescence parameter space.

KOI-227, KOI-319 and KOI-884 are identified here as (at least) two planet systems. For KOI-319 and KOI-884, the observed Transit Timing Variations (TTVs) of the inner transiting planet are used to detect an outer non-transiting planet. The outer planet in KOI-884 is 2.6 Jupiter masses and has the orbital period just narrow of the 3:1 resonance with the inner planet (orbital period ratio 2.93). The distribution of parameters inferred from KOI-319.01's TTVs is bimodal with either a 1.6 Neptune-mass (M_N) planet wide of the 5:3 resonance (period 80.1 d) or a 1 Saturn-mass planet wide of the 7:3 resonance (period 109.2 d). The radial velocity measurements can be used in this case to determine which of these parameter modes is correct. KOI-227.01's TTVs with large 10 hour amplitude can be obtained for planetary-mass companions in various major resonances. Based on the Bayesian evidence, the current TTV data favor the outer 2:1 resonance with a companion mass 1.5 M_N, but this solution implies a very large density of KOI-227.01. The inner and outer 3:2 resonance solutions with sub-Neptune-mass companions are physically more plausible, but will need to be verified., Comment: Astrophysical Journal

The problem of reconstructing the sky position of compact binary coalescences detected via gravitational waves is a central one for future observations with the ground-based network of gravitational-wave laser interferometers, such as Advanced LIGO and Advanced Virgo. Different techniques for sky localisation have been independently developed. They can be divided in two broad categories: fully coherent Bayesian techniques, which are high-latency and aimed at in-depth studies of all the parameters of a source, including sky position, and "triangulation-based" techniques, which exploit the data products from the search stage of the analysis to provide an almost real-time approximation of the posterior probability density function of the sky location of a detection candidate. These techniques have previously been applied to data collected during the last science runs of gravitational-wave detectors operating in the so-called initial configuration. Here, we develop and analyse methods for assessing the self-consistency of parameter estimation methods and carrying out fair comparisons between different algorithms, addressing issues of efficiency and optimality. These methods are general, and can be applied to parameter estimation problems other than sky localisation. We apply these methods to two existing sky localisation techniques representing the two above-mentioned categories, using a set of simulated inspiral-only signals from compact binary systems with total mass $\le 20\,M_\odot$ and non-spinning components. We compare the relative advantages and costs of the two techniques and show that sky location uncertainties are on average a factor $\approx 20$ smaller for fully coherent techniques than for the specific variant of the "triangulation-based" technique used during the last science runs, at the expense of a factor $\approx 1000$ longer processing time.

A new Bayesian software package for the analysis of pulsar timing data is presented in the form of TempoNest which allows for the robust determination of the non-linear pulsar timing solution simultaneously with a range of additional stochastic parameters. This includes both red spin noise and dispersion measure variations using either power law descriptions of the noise, or through a model-independent method that parameterises the power at individual frequencies in the signal. We use TempoNest to show that at noise levels representative of current datasets in the European Pulsar Timing Array (EPTA) and International Pulsar Timing Array (IPTA) the linear timing model can underestimate the uncertainties of the timing solution by up to an order of magnitude. We also show how to perform Bayesian model selection between different sets of timing model and stochastic parameters, for example, by demonstrating that in the pulsar B1937+21 both the dispersion measure variations and spin noise in the data are optimally modelled by simple power laws. Finally we show that not including the stochastic parameters simultaneously with the timing model can lead to unpredictable variation in the estimated uncertainties, compromising the robustness of the scientific results extracted from such analysis., Comment: 21 pages, 12 Figures

We present the first public release of our generic neural network training algorithm, called SkyNet. This efficient and robust machine learning tool is able to train large and deep feed-forward neural networks, including autoencoders, for use in a wide range of supervised and unsupervised learning applications, such as regression, classification, density estimation, clustering and dimensionality reduction. SkyNet uses a `pre-training' method to obtain a set of network parameters that has empirically been shown to be close to a good solution, followed by further optimisation using a regularised variant of Newton's method, where the level of regularisation is determined and adjusted automatically; the latter uses second-order derivative information to improve convergence, but without the need to evaluate or store the full Hessian matrix, by using a fast approximate method to calculate Hessian-vector products. This combination of methods allows for the training of complicated networks that are difficult to optimise using standard backpropagation techniques. SkyNet employs convergence criteria that naturally prevent overfitting, and also includes a fast algorithm for estimating the accuracy of network outputs. The utility and flexibility of SkyNet are demonstrated by application to a number of toy problems, and to astronomical problems focusing on the recovery of structure from blurred and noisy images, the identification of gamma-ray bursters, and the compression and denoising of galaxy images. The SkyNet software, which is implemented in standard ANSI C and fully parallelised using MPI, is available at http://www.mrao.cam.ac.uk/software/skynet/., Comment: 19 pages, 21 figures, 7 tables; this version is re-submission to MNRAS in response to referee comments; software available at http://www.mrao.cam.ac.uk/software/skynet/

GJ667C is the least massive component of a triple star system which lies at a distance of about 6.8 pc (22.1 light-years) from Earth. GJ667C has received much attention recently due to the claims that it hosts up to seven planets including three super-Earths inside the habitable zone. We present a Bayesian technique for the analysis of radial velocity (RV) data-sets in the presence of correlated noise component ("red noise"), with unknown parameters. We also introduce hyper-parameters in our model in order to deal statistically with under or over-estimated error bars on measured RVs as well as inconsistencies between different data-sets. By applying this method to the RV data-set of GJ667C, we show that this data-set contains a significant correlated (red) noise component with correlation timescale for HARPS data of order 9 days. Our analysis shows that the data only provides strong evidence for the presence of two planets: GJ667Cb and c with periods 7.19d and 28.13d respectively, with some hints towards the presence of a third signal with period 91d. The planetary nature of this third signal is not clear and additional RV observations are required for its confirmation. Previous claims of the detection of additional planets in this system are due the erroneous assumption of white noise. Using the standard white noise assumption, our method leads to the detection of up to five signals in this system. We also find that with the red noise model, the measurement uncertainties from HARPS for this system are under-estimated at the level of ~50 per cent., Comment: 10 pages, 6 figures, 3 tables, accepted for publication in MNRAS. Some minor changes in response to referee's comments. arXiv admin note: substantial text overlap with arXiv:1012.5129

We present CARMA observations of a massive galaxy cluster discovered in the AMI blind SZ survey. Without knowledge of the cluster redshift a Bayesian analysis of the AMI, CARMA and joint AMI & CARMA uv-data is used to quantify the detection significance and parameterise both the physical and observational properties of the cluster whilst accounting for the statistics of primary CMB anisotropies, receiver noise and radio sources. The joint analysis of the AMI & CARMA uv-data was performed with two parametric physical cluster models: the {\beta}-model; and the model described in Olamaie et al. 2012 with the pressure profile fixed according to Arnaud et al. 2010. The cluster mass derived from these different models is comparable but our Bayesian evidences indicate a preference for the {\beta}-profile which we, therefore, use throughout our analysis. From the CARMA data alone we obtain a Bayesian probability of detection ratio of 12.8:1 when assuming that a cluster exists within our search area; alternatively assuming that Jenkins et al. 2001 accurately predicts the number of clusters as a function of mass and redshift, the Bayesian probability of detection is 0.29:1. From the analysis of the AMI or AMI & CARMA data the probability of detection ratio exceeds 4.5x10^3:1. Performing a joint analysis of the AMI & CARMA data with a physical cluster model we derive the total mass internal to r200 as MT,200 = 4.1x10^14Msun. Using a phenomenological {\beta}-model to quantify the temperature decrement as a function of angular distance we find a central SZ temperature decrement of 170{\mu}K in the AMI & CARMA data. The SZ decrement in the CARMA data is weaker than expected and we speculate that this is a consequence of the cluster morphology. In a forthcoming study we will assess the impact of cluster morphology on the SZ decrements that are observed with interferometers such as AMI and CARMA., Comment: 13 pages, 13 figures, 6 tables. Accepted by MNRAS

We have obtained deep SZ observations towards 15 of the apparently hottest XMM Cluster Survey (XCS) clusters that can be observed with the Arcminute Microkelvin Imager (AMI). We use a Bayesian analysis to quantify the significance of our SZ detections. We detect the SZ effect at high significance towards three of the clusters and at lower significance for a further two clusters. Towards the remaining ten clusters, no clear SZ signal was measured. We derive cluster parameters using the XCS mass estimates as a prior in our Bayesian analysis. For all AMI-detected clusters, we calculate large-scale mass and temperature estimates while for all undetected clusters we determine upper limits on these parameters. We find that the large- scale mean temperatures derived from our AMI SZ measurements (and the upper limits from null detections) are substantially lower than the XCS-based core-temperature estimates. For clusters detected in the SZ, the mean temperature is, on average, a factor of 1.4 lower than temperatures from the XCS. For clusters undetected in SZ, the average 68% upper limit on the mean temperature is a factor of 1.9 below the XCS temperature., Comment: 18 pages, 14 figures, 6 tables. Accepted by MNRAS

We present 16-GHz Sunyaev-Zel'dovich observations using the Arcminute Microkelvin Imager (AMI) and subsequent Bayesian analysis of six galaxy clusters at redshift ($z \approx 1$) chosen from an X-ray and Infrared selected sample from Culverhouse et al. (2010). In the subsequent analysis we use two cluster models, an isothermal \beta-model and a Dark Matter GNFW (DM-GNFW) model in order to derive a formal detection probability and the cluster parameters. We detect two clusters (CLJ1415+3612 & XMJ0830+5241) and measure their total masses out to a radius of 200 $\times$ the critical density at the respective cluster's redshift. For CLJ1415+3612 and XMJ0830+5241, we find M_{\mathrm{T},200} for each model, which agree with each other for each cluster. We also present maps before and after source subtraction of the entire sample and provide 1D and 2D posterior marginalised probability distributions for each fitted cluster profile parameter of the detected clusters. Using simulations which take into account the measured source environment from the AMI Large Array (LA), source confusion noise, CMB primordials, instrument noise, we estimate from low-radius X-ray data from Culverhouse et al. (2010), the detectability of each cluster in the sample and compare it with the result from the Small Array (SA) data. Furthermore, we discuss the validity of the assumptions of isothermality and constant gas mass fraction. We comment on the bias that these small-radius estimates introduce to large-radius SZ predictions. In addition, we follow-up the two detections with deep, single-pointed LA observations. We find a 3 sigma tentative decrement toward CLJ1415+3612 at high-resolution and a 5 sigma high-resolution decrement towards XMJ0830+5241., Comment: 11 pages, 10 figures, accepted by Monthly Notices of the Royal Astronomical Society

The Arcminute Microkelvin Imager (AMI) is a telescope specifically designed for high sensitivity measurements of low-surface-brightness features at cm-wavelength and has unique, important capabilities. It consists of two interferometer arrays operating over 13.5-18 GHz that image structures on scales of 0.5-10 arcmin with very low systematics. The Small Array (AMI-SA; ten 3.7-m antennas) couples very well to Sunyaev-Zel'dovich features from galaxy clusters and to many Galactic features. The Large Array (AMI-LA; eight 13-m antennas) has a collecting area ten times that of the AMI-SA and longer baselines, crucially allowing the removal of the effects of confusing radio point sources from regions of low surface-brightness, extended emission. Moreover AMI provides fast, deep object surveying and allows monitoring of large numbers of objects. In this White Paper we review the new science - both Galactic and extragalactic - already achieved with AMI and outline the prospects for much more., Comment: 20 pages, 11 figures; white paper. Revised author list, section IB, section IIIC2, references

We present 16-GHz AMI SZ observations of 19 clusters with L_X >7x10^37 W (h50=1) selected from the LoCuS survey (0.142

In this paper we present an algorithm for rapid Bayesian analysis that combines the benefits of nested sampling and artificial neural networks. The blind accelerated multimodal Bayesian inference (BAMBI) algorithm implements the MultiNest package for nested sampling as well as the training of an artificial neural network (NN) to learn the likelihood function. In the case of computationally expensive likelihoods, this allows the substitution of a much more rapid approximation in order to increase significantly the speed of the analysis. We begin by demonstrating, with a few toy examples, the ability of a NN to learn complicated likelihood surfaces. BAMBI's ability to decrease running time for Bayesian inference is then demonstrated in the context of estimating cosmological parameters from Wilkinson Microwave Anisotropy Probe and other observations. We show that valuable speed increases are achieved in addition to obtaining NNs trained on the likelihood functions for the different model and data combinations. These NNs can then be used for an even faster follow-up analysis using the same likelihood and different priors. This is a fully general algorithm that can be applied, without any pre-processing, to other problems with computationally expensive likelihood functions., Comment: 12 pages, 8 tables, 17 figures; accepted by MNRAS; v2 to reflect minor changes in published version

We present follow-up observations of two galaxy clusters detected blindly via the Sunyaev-Zel'dovich (SZ) effect and released in the Planck Early Release Compact Source Catalogue. We use the Arcminute Microkelvin Imager, a dual-array 14-18 GHz radio interferometer. After radio source subtraction, we find a SZ decrement of integrated flux density -1.08+/-0.10 mJy toward PLCKESZ G121.11+57.01, and improve the position measurement of the cluster, finding the centre to be RA 12 59 36.4, Dec +60 04 46.8, to an accuracy of 20 arcseconds. The region of PLCKESZ G115.71+17.52 contains strong extended emission, so we are unable to confirm the presence of this cluster via the SZ effect., Comment: 4 tables, 3 figures, revised after referee's comments and resubmitted to MNRAS

We present an analysis of observations made with the Arcminute Microkelvin Imager (AMI) and the Canada-France-Hawaii Telescope (CFHT) of six galaxy clusters in a redshift range of 0.16--0.41. The cluster gas is modelled using the Sunyaev--Zel'dovich (SZ) data provided by AMI, while the total mass is modelled using the lensing data from the CFHT. In this paper, we: i) find very good agreement between SZ measurements (assuming large-scale virialisation and a gas-fraction prior) and lensing measurements of the total cluster masses out to r_200; ii) perform the first multiple-component weak-lensing analysis of A115; iii) confirm the unusual separation between the gas and mass components in A1914; iv) jointly analyse the SZ and lensing data for the relaxed cluster A611, confirming our use of a simulation-derived mass-temperature relation for parameterizing measurements of the SZ effect., Comment: 22 pages, 12 figures, 12 tables, published by MNRAS

We present observations from the Small Array of the Arcminute Microkelvin Imager (AMI) of eight high X-ray luminosity galaxy cluster systems selected from the Local Cluster Substructure Survey (LoCuSS) sample.We detect the Sunyaev-Zel'dovich (SZ) effect in seven of these clusters. With the assumptions that galaxy clusters are isothermal, have a density profile described by a spherical b -model and obey the theoretical M-T relation, we are able to derive cluster parameters at r200 from our SZ data. With the additional assumption of hydrostatic equilibrium we are able to derive parameters at r500. We present posterior probability distributions for cluster parameters such as mass, radius and temperature (TSZ, MT). Combining our sample with that of AMI Consortium: Rodr'\iguez-Gonz\'alvez et al. (2011) and using large-radius X-ray temperature estimates (TX) from Chandra and Suzaku observations, we find that there is reasonable correspondence between TX and TSZ,MT values at low TX, but that for clusters with TX above around 6keV the correspondence breaks down with TX exceeding TSZ, MT; we stress that this finding is based on just ten clusters., Comment: 17 pages, 4 tables, 14 figures

We present 16-GHz observations using the Arcminute Microkelvin Imager (AMI) of 11 clusters with 7 x 10^{37}W < L_X < 11 x 10^{37}W (h_{50}=1.0) selected from the Local Cluster Substructure Survey (LoCuSS) and compare them to X-ray data. We use a fast, Bayesian cluster analysis to explore the high-dimensional parameter space of the cluster-plus-sources model and obtain robust cluster parameter estimates in the presence of radio point sources, receiver noise and primordial CMB anisotropy. Our analysis fits a spherical, isothermal beta-model to our data and assumes the cluster follows the theoretical mass-temperature relation. Large-scale cluster parameters internal to r_{500} are derived under the assumption of hydrostatic equilibrium. Posterior distributions for the large-scale parameters of 8 of our clusters are given; SZ effects towards Abell 1704 and Zw0857.9+2107 were not detected and our spherical beta-profile was found to be an inadequate fit to the decrement on our map for Abell 2409., Comment: 17 pages, 12 figures, 4 tables This paper has been withdrawn by the authors due to major changes

Most Sunyaev--Zel'dovich (SZ) and X-ray analyses of galaxy clusters try to constrain the cluster total mass and/or gas mass using parameterised models and assumptions of spherical symmetry and hydrostatic equilibrium. By numerically exploring the probability distributions of the cluster parameters given the simulated interferometric SZ data in the context of Bayesian methods, and assuming a beta-model for the electron number density we investigate the capability of this model and analysis to return the simulated cluster input quantities via three rameterisations. In parameterisation I we assume that the T is an input parameter. We find that parameterisation I can hardly constrain the cluster parameters. We then investigate parameterisations II and III in which fg(r200) replaces temperature as a main variable. In parameterisation II we relate M_T(r200) and T assuming hydrostatic equilibrium. We find that parameterisation II can constrain the cluster physical parameters but the temperature estimate is biased low. In parameterisation III, the virial theorem replaces the hydrostatic equilibrium assumption. We find that parameterisation III results in unbiased estimates of the cluster properties. We generate a second simulated cluster using a generalised NFW (GNFW) pressure profile and analyse it with an entropy based model to take into account the temperature gradient in our analysis and improve the cluster gas density distribution. This model also constrains the cluster physical parameters and the results show a radial decline in the gas temperature as expected. The mean cluster total mass estimates are also within 1 sigma from the simulated cluster true values. However, we find that for at least interferometric SZ analysis in practice at the present time, there is no differences in the AMI visibilities between the two models. This may of course change as the instruments improve., Comment: 19 pages, 13 tables, 24 figures

We present 13.9-18.2 GHz observations of the Sunyaev-Zel'dovich (SZ) effect towards Abell 2146 using the Arcminute Microkelvin Imager (AMI). The cluster is detected with a peak SNR ratio of 13 sigma in the radio source subtracted map. Comparison of the SZ and X-ray images suggests that they both have extended regions which lie approximately perpendicular to one another, with their emission peaks significantly displaced. These features indicate non-uniformities in the distributions of the gas temperature and pressure, indicative of a cluster merger. We use a Bayesian cluster analysis to explore the high-dimensional parameter space of the cluster-plus-sources model to obtain cluster parameter estimates in the presence of radio point sources, receiver noise and primordial CMB anisotropy; the probability of SZ + CMB primordial structure + radio sources + receiver noise to CMB + radio sources + receiver noise is 3 x 10^{6}:1. We compare the results from three different cluster models. Our preferred model exploits the observation that the gas fractions do not appear to vary greatly between clusters. Given the relative masses of the two merging systems in Abell 2146, the mean gas temperature can be deduced from the virial theorem (assuming all of the kinetic energy is in the form of internal gas energy) without being affected significantly by the merger event, provided the primary cluster was virialized before the merger. In this model we fit a simple spherical isothermal beta-model, despite the inadequacy of this model for a merging system like Abell 2146, and assume the cluster follows the mass-temperature relation of a virialized, singular, isothermal sphere. We note that this model avoids inferring large-scale cluster parameters internal to r_200 under the widely used assumption of hydrostatic equilibrium. We find that at r_200 M_T= 4.1 \pm 0.5 x 10^{14} h^{-1}M_sun and T=4.5 \pm 0.5 keV., Comment: 14 pages, 15 figures, 7 tables

We present observations using the Small Array of the Arcminute Microkelvin Imager (AMI; 14-18 GHz) of four Abell and three MACS clusters spanning 0.171-0.686 in redshift. We detect Sunyaev-Zel'dovich (SZ) signals in five of these without any attempt at source subtraction, although strong source contamination is present. With radio-source measurements from high-resolution observations, and under the assumptions of spherical $\beta$-model, isothermality and hydrostatic equilibrium, a Bayesian analysis of the data in the visibility plane detects extended SZ decrements in all seven clusters over and above receiver noise, radio sources and primary CMB imprints. Bayesian evidence ratios range from 10^{11}:1 to 10^{43}:1 for six of the clusters and 3000:1 for one with substantially less data than the others. We present posterior probability distributions for, e.g., total mass and gas fraction averaged over radii internal to which the mean overdensity is 1000, 500 and 200, r_200 being the virial radius. Reaching r_200 involves some extrapolation for the nearer clusters but not for the more-distant ones. We find that our estimates of gas fraction are low (compared with most in the literature) and decrease with increasing radius. These results appear to be consistent with the notion that gas temperature in fact falls with distance (away from near the cluster centre) out to the virial radius., Comment: 18 pages, 10 figures, submitted to MNRAS (updated authors and fixed Figure 1)

Coalescing massive Black Hole binaries are the strongest and probably the most important gravitational wave sources in the LISA band. The spin and orbital precessions bring complexity in the waveform and make the likelihood surface richer in structure as compared to the non-spinning case. We introduce an extended multimodal genetic algorithm which utilizes the properties of the signal and the detector response function to analyze the data from the third round of mock LISA data challenge (MLDC 3.2). The performance of this method is comparable, if not better, to already existing algorithms. We have found all five sources present in MLDC 3.2 and recovered the coalescence time, chirp mass, mass ratio and sky location with reasonable accuracy. As for the orbital angular momentum and two spins of the Black Holes, we have found a large number of widely separated modes in the parameter space with similar maximum likelihood values., Comment: 25 pages, 9 figures

The Mock LISA Data Challenges are a program to demonstrate LISA data-analysis capabilities and to encourage their development. Each round of challenges consists of one or more datasets containing simulated instrument noise and gravitational waves from sources of undisclosed parameters. Participants analyze the datasets and report best-fit solutions for the source parameters. Here we present the results of the third challenge, issued in Apr 2008, which demonstrated the positive recovery of signals from chirping Galactic binaries, from spinning supermassive--black-hole binaries (with optimal SNRs between ~ 10 and 2000), from simultaneous extreme-mass-ratio inspirals (SNRs of 10-50), from cosmic-string-cusp bursts (SNRs of 10-100), and from a relatively loud isotropic background with Omega_gw(f) ~ 10^-11, slightly below the LISA instrument noise., Comment: 12 pages, 2 figures, proceedings of the 8th Edoardo Amaldi Conference on Gravitational Waves, New York, June 21-26, 2009

We consider the problem of characterisation of burst sources detected with the Laser Interferometer Space Antenna (LISA) using the multi-modal nested sampling algorithm, MultiNest. We use MultiNest as a tool to search for modelled bursts from cosmic string cusps, and compute the Bayesian evidence associated with the cosmic string model. As an alternative burst model, we consider sine-Gaussian burst signals, and show how the evidence ratio can be used to choose between these two alternatives. We present results from an application of MultiNest to the last round of the Mock LISA Data Challenge, in which we were able to successfully detect and characterise all three of the cosmic string burst sources present in the release data set. We also present results of independent trials and show that MultiNest can detect cosmic string signals with signal-to-noise ratio (SNR) as low as ~7 and sine-Gaussian signals with SNR as low as ~8. In both cases, we show that the threshold at which the sources become detectable coincides with the SNR at which the evidence ratio begins to favour the correct model over the alternative., Comment: 21 pages, 11 figures, accepted by CQG; v2 has minor changes for consistency with accepted version

Using the Arcminute Microkelvin Imager (AMI) at 16 GHz and the Very Small Array (VSA) at 33 GHz to make follow-up observations of sources in the New Extragalactic WMAP Point Source Catalogue, we have investigated the flux-density variability in a complete sample of 97 sources over timescales of a few months to approximately 1.5 years. We find that 53 per cent of the 93 sources, for which we have multiple observations, are variable, at the 99 per cent confidence level, above the flux density calibration uncertainties of approximately 4 per cent at 16 GHz; the fraction of sources having varied by more than 20 per cent is 15 per cent at 16 GHz and 20 per cent at 33 GHz. Not only is this common occurrence of variability at high frequency of interest for source physics, but strategies for coping with source contamination in CMB work must take this variability into account. There is no strong evidence of a correlation between variability and flux density for the sample as a whole. Using a maximum-likelihood method, we calculate the correlation in the variability at the two frequencies in a subset of sources classified as variable from both the AMI and VSA data and find the Pearson product-moment correlation coefficient to be very high (0.955 +/- 0.034). We also find the degree of variability at 16 GHz (0.202 +/- 0.028) to be very similar to that at 33 GHz (0.224 +/- 0.039). Finally, we have investigated the relationship between variability and spectral index, alpha_13.9^33.75 (where S proportional to nu^-alpha), and find a significant difference in the spectral indices of the variable sources (-0.06 +/- 0.05) and non-variable sources (0.13 +/- 0.04)., Comment: 12 pages and 10 figures

We present follow-up observations of 97 point sources from the Wilkinson Microwave Anisotropy Probe (WMAP) 3-year data, contained within the New Extragalactic WMAP Point Source (NEWPS) catalogue between declinations of -4 and +60 degrees; the sources form a flux-density-limited sample complete to 1.1 Jy (approximately 5 sigma) at 33 GHz. Our observations were made at 16 GHz using the Arcminute Microkelvin Imager (AMI) and at 33 GHz with the Very Small Array (VSA). 94 of the sources have reliable, simultaneous -- typically a few minutes apart -- observations with both telescopes. The spectra between 13.9 and 33.75 GHz are very different from those of bright sources at low frequency: 44 per cent have rising spectra (alpha < 0.0), where flux density is proportional to frequency^-alpha, and 93 per cent have spectra with alpha < 0.5; the median spectral index is 0.04. For the brighter sources, the agreement between VSA and WMAP 33-GHz flux densities averaged over sources is very good. However, for the fainter sources, the VSA tends to measure lower values for the flux densities than WMAP. We suggest that the main cause of this effect is Eddington bias arising from variability., Comment: 12 pages, 13 figures, submitted to MNRAS

We describe an application of the MultiNest algorithm to gravitational wave data analysis. MultiNest is a multimodal nested sampling algorithm designed to efficiently evaluate the Bayesian evidence and return posterior probability densities for likelihood surfaces containing multiple secondary modes. The algorithm employs a set of live points which are updated by partitioning the set into multiple overlapping ellipsoids and sampling uniformly from within them. This set of live points climbs up the likelihood surface through nested iso-likelihood contours and the evidence and posterior distributions can be recovered from the point set evolution. The algorithm is model-independent in the sense that the specific problem being tackled enters only through the likelihood computation, and does not change how the live point set is updated. In this paper, we consider the use of the algorithm for gravitational wave data analysis by searching a simulated LISA data set containing two non-spinning supermassive black hole binary signals. The algorithm is able to rapidly identify all the modes of the solution and recover the true parameters of the sources to high precision., Comment: 18 pages, 4 figures, submitted to Class. Quantum Grav; v2 includes various changes in light of referee's comments

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 comments