Nested Sampling Algorithm Research Articles

THE HUNT FOR EXOMOONS WITHKEPLER(HEK). I. DESCRIPTION OF A NEW OBSERVATIONAL PROJECT

Two decades ago, empirical evidence concerning the existence and frequency of planets around stars, other than our own, was absent. Since this time, the detection of extrasolar planets from Jupiter-sized to most recently Earth-sized worlds has blossomed and we are finally able to shed light on the plurality of Earth-like, habitable planets in the cosmos. Extrasolar moons may also be frequent habitable worlds but their detection or even systematic pursuit remains lacking in the current literature. Here, we present a description of the first systematic search for extrasolar moons as part of a new observational project called "The Hunt for Exomoons with Kepler" (HEK). The HEK project distills the entire list of known transiting planet candidates found by Kepler (2326 at the time of writing) down to the most promising candidates for hosting a moon. Selected targets are fitted using a multimodal nested sampling algorithm coupled with a planet-with-moon light curve modelling routine. By comparing the Bayesian evidence of a planet-only model to that of a planet-with-moon, the detection process is handled in a Bayesian framework. In the case of null detections, upper limits derived from posteriors marginalised over the entire prior volume will be provided to inform the frequency of large moons around viable planetary hosts, eta-moon. After discussing our methodologies for target selection, modelling, fitting and vetting, we provide two example analyses.

Bayesian model-based filter design in acoustical signal processing applications

Digital filter design finds applications across many disciplines. Acoustic applications include, but are not limited to, impedance boundary conditions in time-domain wave-based room-acoustics modeling, head-related transfer functions, and loudspeaker equalization. The process of filter design can be approached in a variety of ways. To design a non-standard filter, optimization methods may be used to solve for the coefficients that minimize the difference from a specified transfer function of a prescribed order. Interpreted as a model-based inference problem, a Bayesian framework, realized through the nested sampling algorithm, provides simultaneous optimal coefficient estimates and a filter order selection criterion. The selection criterion implicitly favors simpler models over more complex models, in this case lower-order filters over higher-order filters. Many acoustical applications are well suited to this type design, where low filter order is important, and the desired system response is unattainable...

Exploring the Energy Landscapes of Protein Folding Simulations with Bayesian Computation

Nested sampling is a Bayesian sampling technique developed to explore probability distributions localized in an exponentially small area of the parameter space. The algorithm provides both posterior samples and an estimate of the evidence (marginal likelihood) of the model. The nested sampling algorithm also provides an efficient way to calculate free energies and the expectation value of thermodynamic observables at any temperature, through a simple post processing of the output. Previous applications of the algorithm have yielded large efficiency gains over other sampling techniques, including parallel tempering. In this article, we describe a parallel implementation of the nested sampling algorithm and its application to the problem of protein folding in a Gō-like force field of empirical potentials that were designed to stabilize secondary structure elements in room-temperature simulations. We demonstrate the method by conducting folding simulations on a number of small proteins that are commonly used for testing protein-folding procedures. A topological analysis of the posterior samples is performed to produce energy landscape charts, which give a high-level description of the potential energy surface for the protein folding simulations. These charts provide qualitative insights into both the folding process and the nature of the model and force field used.

Exploring the Energy Landscapes of Protein Folding Simulations with Bayesian Computation

Nested sampling is a Bayesian sampling technique developed to explore probability distributions localised in an exponentially small area of the parameter space. The algorithm produces both posterior samples and an estimate of the evidence of the model. The nested sampling algorithm also provides an efficient way to calculate free energies and the expectation value of thermodynamic observables at any temperature, through a simple post-processing of the output. Previous applications of the algorithm have yielded large efficiency gains over other sampling techniques.

Model selection in finite element model updating using the Bayesian evidence statistic

This paper considers the problem of finite element model (FEM) updating in the context of model selection. The FEM updating problem arises from the need to update the initial FE model that does not match the measured real system outputs. This inverse system identification-problem is made even more complex by the uncertainties in modeling some of the structural parameters. Such uncertainty often results in a number of competing forms of FE models being proposed which leads to lack of consensus in the field. A model can be formulated in a number of ways; by the number, the location and the form of the updating parameters. We propose the use of a Bayesian evidence statistic to help decide on the best model from any given set of models. This statistic uses the recently developed stochastic nested sampling algorithm whose by-product is the posterior samples of the updated model parameters. Two examples of real structures are each modeled by a number of competing finite element models. The individual model evidences are compared using the Bayes factor, which is the ratio of evidences. Jeffrey's scale is then used to determine the significance of the model differences obtained through the Bayes factor.

Bayesian model selection for electromagnetic kaon production on the nucleon

We present the results of a Bayesian analysis of a Regge model to describe the background contribution for K+Λ and K+Σ0 photoproduction. The model is based on the exchange of K+(494) and K*+(892) trajectories in the t-channel. We utilise the Bayesian evidence Z to determine the best model variant for each channel. The Bayesian evidence integrals were calculated using the Nested Sampling algorithm. For different prior widths, we find decisive Bayesian evidence (ΔlnZ≈24) for a K+Λ photoproduction Regge model with a positive vector coupling and a negative tensor coupling constant for the K*+(892) trajectory, and a rotating phase factor for both trajectories. Using the χ2 minimisation method, one could not draw this conclusion from the same dataset. For the K+Σ0 photoproduction Regge model, on the other hand, the difference between the evidence integrals is insufficient to pinpoint one model variant.

Efficient Sampling of Atomic Configurational Spaces

We describe a method to explore the configurational phase space of chemical systems. It is based on the nested sampling algorithm recently proposed by Skilling (AIP Conf. Proc. 2004, 395; J. Bayesian Anal. 2006, 1, 833) and allows us to explore the entire potential energy surface (PES) efficiently in an unbiased way. The algorithm has two parameters which directly control the trade-off between the resolution with which the space is explored and the computational cost. We demonstrate the use of nested sampling on Lennard-Jones (LJ) clusters. Nested sampling provides a straightforward approximation for the partition function; thus, evaluating expectation values of arbitrary smooth operators at arbitrary temperatures becomes a simple postprocessing step. Access to absolute free energies allows us to determine the temperature-density phase diagram for LJ cluster stability. Even for relatively small clusters, the efficiency gain over parallel tempering in calculating the heat capacity is an order of magnitude or more. Furthermore, by analyzing the topology of the resulting samples, we are able to visualize the PES in a new and illuminating way. We identify a discretely valued order parameter with basins and suprabasins of the PES, allowing a straightforward and unambiguous definition of macroscopic states of an atomistic system and the evaluation of the associated free energies.

Nested sampling as a tool for LISA data analysis

Nested sampling is a technique for efficiently computing the probability of a data set under a particular hypothesis, also called the Bayesian Evidence or Marginal Likelihood, and for evaluating the posterior. MULTINEST is a multi-modal nested sampling algorithm which has been designed to efficiently explore and characterize posterior probability surfaces containing multiple secondary solutions. We have applied the MULTINEST algorithm to a number of problems in gravitational wave data analysis. In this article, we describe the algorithm and present results for several applications of the algorithm to analysis of mock LISA data. We summarise recently published results for a test case in which we searched for two non-spinning black hole binary merger signals in simulated LISA data. We also describe results obtained with MULTINEST in the most recent round of the Mock LISA Data Challenge (MLDC), in which the algorithm was used to search for and characterise both spinning supermassive black hole binary inspirals and bursts from cosmic string cusps. In all these applications, the algorithm found the correct number of signals and efficiently recovered the posterior probability distribution. Moreover, in most cases the waveform corresponding to the best a-posteriori parameters had an overlap in excess of 99% with the true signal.

Classifying LISA gravitational wave burst signals using Bayesian evidence

We consider the problem of characterization of burst sources detected by 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 characterize 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.

Use of the MULTINEST algorithm for gravitational wave data analysis

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.

Cold dark matter isocurvature perturbations: Constraints and model selection

We use cosmic microwave background radiation (WMAP and ACBAR), large-scale structure (SDSS luminous red galaxies), and supernova (SNLS) data to constrain the possible contribution of cold dark matter isocurvature modes to the primordial perturbation spectrum. We consider three different admixtures with adiabatic modes in a flat {lambda}CDM cosmology with no tensor modes: fixed correlations with a single spectral index; general correlations with a single spectral index; and general correlations with independent spectral indices for each mode. For fixed correlations, we verify the WMAP analysis for fully uncorrelated and anticorrelated modes, while for general correlations with a single index we find a small tightening of the constraint on the fractional contribution of isocurvature modes to the observed power over earlier work. For generally correlated modes and independent spectral indices our results are quite different to previous work, needing a doubling of prior space for the isocurvature spectral index in order to explore adequately the region of high likelihood. Standard Markov-Chain Monte Carlo techniques proved to be inadequate for this particular application; instead, our results are obtained with nested sampling. We also use the Bayesian evidence, calculated simply in the nested-sampling algorithm, to compare models, finding the pure adiabatic model to bemore » favored over all our isocurvature models. This favoring is such that the logarithm of the Bayes factor, lnB<-2 for all models and lnB<-5 in the cases of fully anticorrelated modes with a single spectral index (the curvaton scenario) and generally correlated modes with a single spectral index.« less

Bayesian inference on the Numerical INJection Analysis (NINJA) data set using a nested sampling algorithm

We present the results of an analysis of the first data release of the Numerical INJection Analysis (NINJA) project to search for gravitational waves generated by the coalescence of black hole binaries. We adopt a Bayesian approach implemented through a nested sampling algorithm to explore how different waveform families affect the confidence of detection of signals generated by numerically evolving a two-body system in full general relativity. In particular we consider the standard second-order post-Newtoninan approximation to the inspiral phase (TaylorF2 approximant) and the inspiral-merger-rigdown phenomenological waveforms suggested by Ajith and collaborators (2008 Phys. Rev. D 77 104017) (IMRPhenomA approximant). We show that the latter family consistently out-performs TaylorF2 waveforms, and 112 of the 126 signals present in the data set are recovered with a log Bayes factor larger than 3. We also carry out a study of the recovered source parameters, and in particular we discuss the errors affecting the determination of the total mass and sky position.

MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics

We present further development and the first public release of our multimodal nested sampling algorithm, called MultiNest. This Bayesian inference tool calculates the evidence, with an associated error estimate, and produces posterior samples from distributions that may contain multiple modes and pronounced (curving) degeneracies in high dimensions. The developments presented here lead to further substantial improvements in sampling efficiency and robustness, as compared to the original algorithm presented in Feroz & Hobson (2008), which itself significantly outperformed existing MCMC techniques in a wide range of astrophysical inference problems. The accuracy and economy of the MultiNest algorithm is demonstrated by application to two toy problems and to a cosmological inference problem focussing on the extension of the vanilla $\Lambda$CDM model to include spatial curvature and a varying equation of state for dark energy. The MultiNest software, which is fully parallelized using MPI and includes an interface to CosmoMC, is available at http://www.mrao.cam.ac.uk/software/multinest/. It will also be released as part of the SuperBayeS package, for the analysis of supersymmetric theories of particle physics, at http://www.superbayes.org

Bayesian model selection applied to room‐acoustic energy decay analysis

The previous work of Xiang et al. [Xiang & Goggans, J. Acoust. Soc. Am. 110 (2001), pp. 1415‐1424; 113 (2003), pp. 2685‐2697; Xiang & Jasa, J. Acoust. Soc. Am. 120 (2006) pp.3744‐3749] successfully applied the Bayesian formulism to estimate multiple decay parameters from Schroeder decay functions measured or calculated in acoustically coupled spaces. In this work we consider a more difficult problem of determining the correct decay model in the presence of energy decay data within the Bayesian framework. We will compare the Annealing/Thermodynamic algorithm [Neal, R. M., Statistics and Computing 11 (2001), pp. 125‐139], the Nested Sampling algorithm [Silvia & Skilling, Data Analysis: A Bayesian Tutorial, Oxford Science Publications (2006)], and a combined Variational/Markov Chain Monte Carlo approach in order to determine the correct decay model. The advantages/shortcomings of these methods are discussed in the context of the decay model selection and parameter estimation using experimentally measured data from real coupled spaces.

A Nested Sampling Algorithm for Cosmological Model Selection

The abundance of cosmological data becoming available means that a wider range of cosmological models are testable than ever before. However, an important distinction must be made between parameter fitting and model selection. While parameter fitting simply determines how well a model fits the data, model selection statistics, such as the Bayesian evidence, are now necessary to choose between these different models, and in particular to assess the need for new parameters. We implement a new evidence algorithm known as nested sampling, which combines accuracy, generality of application, and computational feasibility, and we apply it to some cosmological data sets and models. We find that a five-parameter model with a Harrison-Zel'dovich initial spectrum is currently preferred.