Showing papers on "Nested sampling algorithm published in 2010"

Bayesian coherent analysis of in-spiral gravitational wave signals with a detector network

Abstract: The present operation of the ground-based network of gravitational-wave laser interferometers in enhanced configuration and the beginning of the construction of second-generation (or advanced) interferometers with planned observation runs beginning by 2015 bring the search for gravitational waves into a regime where detection is highly plausible. The development of techniques that allow us to discriminate a signal of astrophysical origin from instrumental artefacts in the interferometer data and to extract the full range of information are therefore some of the primary goals of the current work. Here we report the details of a Bayesian approach to the problem of inference for gravitational wave observations using a network (containing an arbitrary number) of instruments, for the computation of the Bayes factor between two hypotheses and the evaluation of the marginalized posterior density functions of the unknown model parameters. The numerical algorithm to tackle the notoriously difficult problem of the evaluation of large multidimensional integrals is based on a technique known as nested sampling, which provides an attractive (and possibly superior) alternative to more traditional Markov-chain Monte Carlo methods. We discuss the details of the implementation of this algorithm and its performance against a Gaussian model of the background noise, considering the specific case of the signal produced by the in-spiral of binary systems of black holes and/or neutron stars, although the method is completely general and can be applied to other classes of sources. We also demonstrate the utility of this approach by introducing a new coherence test to distinguish between the presence of a coherent signal of astrophysical origin in the data of multiple instruments and the presence of incoherent accidental artefacts, and the effects on the estimation of the source parameters as a function of the number of instruments in the network.

Properties of nested sampling

Abstract: Le lien donne acces a la version document de travail intitulee "Contemplating Evidence: properties, extensions of, and alternatives to Nested Sampling"

Efficient Sampling of Atomic Configurational Spaces

Abstract: 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.

Bayesian model comparison in cosmology with Population Monte Carlo

Abstract: We use Bayesian model selection techniques to test extensions of the standard flat Λ cold dark matter (ΛCDM) paradigm. Dark-energy and curvature scenarios, and primordial perturbation models are considered. To that end, we calculate the Bayesian evidence in favour of each model using Population Monte Carlo (PMC), a new adaptive sampling technique which was recently applied in a cosmological context. In contrast to the case of other sampling-based inference techniques such as Markov chain Monte Carlo (MCMC), the Bayesian evidence is immediately available from the PMC sample used for parameter estimation without further computational effort, and it comes with an associated error evaluation. Also, it provides an unbiased estimator of the evidence after any fixed number of iterations and it is naturally parallelizable, in contrast with MCMC and nested sampling methods. By comparison with analytical predictions for simulated data, we show that our results obtained with PMC are reliable and robust. The variability in the evidence evaluation and the stability for various cases are estimated both from simulations and from data. For the cases we consider, the log-evidence is calculated with a precision of better than 0.08. Using a combined set of recent cosmic microwave background, type Ia supernovae and baryonic acoustic oscillation data, we find inconclusive evidence between flat ΛCDM and simple dark-energy models. A curved universe is moderately to strongly disfavoured with respect to a flat cosmology. Using physically well-motivated priors within the slow-roll approximation of inflation, we find a weak preference for a running spectral index. A Harrison–Zel'dovich spectrum is weakly disfavoured. With the current data, tensor modes are not detected; the large prior volume on the tensor-to-scalar ratio r results in moderate evidence in favour of r= 0.

Exploring the Energy Landscapes of Protein Folding Simulations with Bayesian Computation

Abstract: Nested sampling is a Bayesian sampling technique developed to explore probability distributions lo- calised 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 algo- rithm 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 (replica exchange). In this paper we describe a parallel implementation of the nested sampling algorithm and its application to the problem of protein folding in a Go-type 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 which are commonly used for testing protein folding procedures: protein G, the SH3 domain of Src tyrosine kinase and chymotrypsin inhibitor 2. 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.

Statistical coverage for supersymmetric parameter estimation: a case study with direct detection of dark matter

Abstract: Models of weak-scale supersymmetry offer viable dark matter (DM) candidates. Their parameter spaces are however rather large and complex, such that pinning down the actual parameter values from experimental data can depend strongly on the employed statistical framework and scanning algorithm. In frequentist parameter estimation, a central requirement for properly constructed confidence intervals is that they cover true parameter values, preferably at exactly the stated confidence level when experiments are repeated infinitely many times. Since most widely-used scanning techniques are optimised for Bayesian statistics, one needs to assess their abilities in providing correct confidence intervals in terms of the statistical coverage. Here we investigate this for the Constrained Minimal Supersymmetric Standard Model (CMSSM) when only constrained by data from direct searches for dark matter. We construct confidence intervals from one-dimensional profile likelihoods and study the coverage by generating several pseudo-experiments for a few benchmark sets of pseudo-true parameters. We use nested sampling to scan the parameter space and evaluate the coverage for the benchmarks when either flat or logarithmic priors are imposed on gaugino and scalar mass parameters. The sampling algorithm has been used in the configuration usually adopted for exploration of the Bayesian posterior. We observe both under- and over-coverage, which in some cases vary quite dramatically when benchmarks or priors are modified. We show how most of the variation can be explained as the impact of explicit priors as well as sampling effects, where the latter are indirectly imposed by physicality conditions. For comparison, we also evaluate the coverage for Bayesian credible intervals, and observe significant under-coverage in those cases.

Nested Sampling as a tool for LISA data analysis

Abstract: 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.

Bayesian model comparison in cosmology with Population

Abstract: We use Bayesian model selection techniques to test extensions of the standard at CDM paradigm. Dark-energy and curvature scenarios, and primordial perturbation models are considered. To that end, we calculate the Bayesian evidence in favour of each model using Population Monte Carlo (PMC), a new adaptive sampling technique which was recently applied in a cosmological context. In contrast to the case of other sampling-based inference techniques such as Markov chain Monte Carlo (MCMC), the Bayesian evidence is immediately available from the PMC sample used for parameter estimation without further computational eort, and it comes with an associated error evaluation. Also, it provides an unbiased estimator of the evidence after any xed number of iterations and it is naturally parallelizable, in contrast with MCMC and nested sampling methods. By comparison with analytical predictions for simulated data, we show that our results obtained with PMC are reliable and robust. The variability in the evidence evaluation and the stability for various cases are estimated both from simulations and from data. For the cases we consider, the log-evidence is calculated with a precision of better than 0:08. Using a combined set of recent CMB, SNIa and BAO data, we nd inconclusive evidence between at

Comment on "Bayesian evidence: can we beat MultiNest using traditional MCMC methods", by Rutger van Haasteren (arXiv:0911.2150)

Abstract: In arXiv:0911.2150, Rutger van Haasteren seeks to criticize the nested sampling algorithm for Bayesian data analysis in general and its M ULTINEST implementation in particular. He introduces a new method for evidence evaluation based on the idea of Voronoi tessellation and requiring samples from the posterior distribution obtained through MCMC based methods. He compares its accuracy and efficiency with M ULTINEST, concluding that it outperforms MULTINEST in several cases. This comparison is completely unfair since the proposed method can not perform the complete Bayesian data analysis including posterior exploration and evidence evaluation on its own while MULTINEST allows one to perform Bayesian data analysis end to end. Furthermore, their criticism of nested sampling (and in turn MULTINEST) is based on a few conceptual misunderstandings of the algorithm. Here we seek to set the record straight.

Entropy-Based Search Algorithm for Experimental Design

Abstract: The scientific method relies on the iterated processes of inference and inquiry. The inference phase consists of selecting the most probable models based on the available data; whereas the inquiry phase consists of using what is known about the models to select the most relevant experiment. Optimizing inquiry involves searching the parameterized space of experiments to select the experiment that promises, on average, to be maximally informative. In the case where it is important to learn about each of the model parameters, the relevance of an experiment is quantified by Shannon entropy of the distribution of experimental outcomes predicted by a probable set of models. If the set of potential experiments is described by many parameters, we must search this high-dimensional entropy space. Brute force search methods will be slow and computationally expensive. We present an entropy-based search algorithm, called nested entropy sampling, to select the most informative experiment for efficient experimental design. This algorithm is inspired by Skilling's nested sampling algorithm used in inference and borrows the concept of a rising threshold while a set of experiment samples are maintained. We demonstrate that this algorithm not only selects highly relevant experiments, but also is more efficient than brute force search. Such entropic search techniques promise to greatly benefit autonomous experimental design.

Bayesian model comparison in cosmology with Population Monte Carlo

Abstract: We use Bayesian model selection techniques to test extensions of the standard flat LambdaCDM paradigm. Dark-energy and curvature scenarios, and primordial perturbation models are considered. To that end, we calculate the Bayesian evidence in favour of each model using Population Monte Carlo (PMC), a new adaptive sampling technique which was recently applied in a cosmological context. The Bayesian evidence is immediately available from the PMC sample used for parameter estimation without further computational effort, and it comes with an associated error evaluation. Besides, it provides an unbiased estimator of the evidence after any fixed number of iterations and it is naturally parallelizable, in contrast with MCMC and nested sampling methods. By comparison with analytical predictions for simulated data, we show that our results obtained with PMC are reliable and robust. The variability in the evidence evaluation and the stability for various cases are estimated both from simulations and from data. For the cases we consider, the log-evidence is calculated with a precision of better than 0.08. Using a combined set of recent CMB, SNIa and BAO data, we find inconclusive evidence between flat LambdaCDM and simple dark-energy models. A curved Universe is moderately to strongly disfavoured with respect to a flat cosmology. Using physically well-motivated priors within the slow-roll approximation of inflation, we find a weak preference for a running spectral index. A Harrison-Zel'dovich spectrum is weakly disfavoured. With the current data, tensor modes are not detected; the large prior volume on the tensor-to-scalar ratio r results in moderate evidence in favour of r=0.

Parameter Fitting of Cosmological Models using Evolutionary Strategies

Abstract: The current tools used for approximating cosmological parameters are Markov Chain Monte Carlo (MCMC) utilities using Metropolis-Hastings and Nested Sampling as the main sampling methods. These tend to have low sampling efficiency as many samples are wasted in trying to find good proposals of points at high regions of likelihood. In the data-rich era in which cosmology is entering, imaging the entire sky and generating PetaBytes of data, this is unsatisfactory. In this project we propose Evolutionary Strategies as a better sampling method. The algorithm works by mutating populations of samples across iterations of the algorithm in order to adapt its behaviour over the current sample’s location in parameter space. The developed method, ESNested, integrates Nested Sampling with Evolutionary Strategies, such that the method is able to approximate the Bayesian Evidence of a distribution. Results show that ESNested returns good parameter fits with a high sampling efficiency. Other methods tend to make a Gaussian assumption on the search space, that can generate a bad parameter fit if the data does not follow this distribution in real life. ESNested does not make this assumption and therefore works well in most search spaces. Using the developed method, various parameter fits, using different combinations of parameters are tested on Cosmological models. By analysing parameter fits, confidence intervals and the importance and correlations of parameters in Cosmological models, this project also looks at what theories in Cosmology do generated posterior samples support or reject.

Bayes Factors via MCMC and Path Sampling - The case of Factor Models

Abstract: We prove a theorem justifying the regularity conditions which are needed for Path Sampling in Factor Models. We then show that the remaining ingredient, namely, MCMC for calculating the integrand at each point in the path, may be seriously flawed, leading to wrong estimates of Bayes factors. We provide a new method of Path Sampling (with Small Change) that works much better than standard Path Sampling in the sense of estimating the Bayes factor better and choosing the correct model more often. When the more complex factor model is true, PS-SC is substantially more accurate. New MCMC diagnostics is provided for these problems in support of our conclusions and recommendations. Some of our ideas for diagnostics and improvement in computation through small changes should apply to other methods of computation of the Bayes factor for model selection.

Monte carlo simulations of the nested fixed-point algorithm

Abstract: There have been substantial advances in dynamic structural models and in the econometric literature about techniques to estimate those models over the past two decades. One area in which these new developments has lagged is in studying robustness to distributional assumptions and finite sample properties in small samples. This paper extends our understanding of the behavior of these estimation techniques by replicating John Rust’s (1987) influential paper using the nested fixed-point algorithm (NFXP) and then using Monte Carlo techniques to examine the finite sample properties of the estimator. I then examine the consequences of the distributional assumptions needed to estimate the model on the parameter estimates. I find that even in sample sizes of up to 8,000 observations, the NFXP can display finite sample bias and variances substantially larger than the theoretical asymptotic variance. This is also true with departures from distributional assumptions, with the mean square error increasing by a factor of 10 for some distributions of unobserved variables. Erik P. Johnson Georgia Institute of Technology 221 Bobby Dodd Way, Atlanta, GA 30332 erik.johnson@econ.gatech.edu Monte Carlo Simulations of the Nested Fixed-Point Algorithm Erik Johnson October 2010 Preliminary and incomplete. Please do not cite or quote.

Computational Methods in Bayesian Statistics

Abstract: This paper focuses on utilizing two different Bayesian methods to deal with a variety of toy problems which occur in data analysis. In particular we implement the Variational Bayesian and Nested Sampling methods to tackle the problems of polynomial selection and Gaussian Mixture Models, comparing the algorithms in terms of processing speed and accuracy. In the problems tackled here it is the Variational Bayesian algorithms which are the faster though both results give similar results.