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 M ulti N est . 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, which itself significantly outperformed existing Markov chain Monte Carlo techniques in a wide range of astrophysical inference problems. The accuracy and economy of the M ulti N est algorithm are demonstrated by application to two toy problems and to a cosmological inference problem focusing on the extension of the vanilla Λ cold dark matter model to include spatial curvature and a varying equation of state for dark energy. The M ulti N est software, which is fully parallelized using MPI and includes an interface to C osmo MC, 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 .     

Limitations of Bayesian Evidence applied to cosmology

There has been increasing interest by cosmologists in applying Bayesian techniques, such as Bayesian Evidence, for model selection. A typical example is in assessing whether observational data favour a cosmological constant over evolving dark energy. In this paper, the example of dark energy is used to illustrate limitations in the application of Bayesian Evidence associated with subjective judgements concerning the choice of model and priors. An analysis of recent cosmological data shows a statistically insignificant preference for a cosmological constant over simple dynamical models of dark energy. It is argued that for nested problems, as considered here, Bayesian parameter estimation can be more informative than computing Bayesian Evidence for poorly motivated physical models.     

Fisher matrix decomposition for dark energy prediction

Within the context of constraining an expansion of the dark energy equation of state   w ( z ),  we show that the eigendecomposition of Fisher matrices is sensitive to both the maximum order of the expansion and the basis set choice. We investigate the Fisher matrix formalism in the case that a particular function is expanded in some basis set. As an example we show results for an all-sky weak lensing tomographic experiment. We show that the set of eigenfunctions is not unique and that the best constrained functions are only reproduced accurately at very higher order   N ≳ 100  , a top-hat basis set requires an even higher order. We show that the common approach used for finding the marginalized eigenfunction errors is sensitive to the choice of  non- w ( z )  parameters and priors. The eigendecomposition of Fisher matrices is a potentially useful tool that can be used to determine the predicted accuracy with which an experiment could constrain   w ( z )  . It also allows for the reconstruction of the redshift sensitivity of the experiment to changes in   w ( z )  . However, the technique is sensitive to both the order and the basis set choice. Publicly available code is available as part of icosmo at http://www.icosmo.org .     

A regularized free-form estimator for dark energy

We construct a simple, regularized estimator for the dark energy equation of state by using the recently introduced linear response approximation. We show that even a simple regularization substantially improves the performance of the free-form fitting approach. The use of the linear response approximation allows an analytical construction of the maximum likelihood estimator, in a convenient and easy to use matrix form. We show that, in principle, such regularized free-form fitting can give us an unbiased estimate of the functional form of the equation of state of dark energy. We show the efficacy of this approach on simulated SuperNova Acceleration Probe class data, but it is easy to generalize this method to include other cosmological tests. We provide a possible explanation for the sweet spots seen in other reconstruction methods.     

A modified χ2-test for cosmic microwave background analyses

We present a new general procedure for determining a given set of quantities. To this end, we define a certain statistic, which we call 'modified  χ²' (χ²_M)  , because of its similarity to the standard  χ²  . The terms of this  χ²_M  are made up of the fluctuations of an unbiased estimator of some statistical quantities and certain weights. Only the diagonal terms of the covariance matrix appear explicitly in our statistic, while the full covariance matrix (and not its inverse) is included implicitly in the calculation of the weights. Choosing these weights, we may obtain, through minimizing  χ²_M  , the estimator that provides the minimum rms, either for those quantities or for the parameters on which these quantities depend. In this paper, we describe our method in the context of cosmic microwave background experiments, in order to obtain either the statistical properties of the maps or the cosmological parameters. The test here is constructed out of some estimator of the two-point correlation function at different angles. For the problem of one-parameter estimation, we show that our method has the same power as the maximum-likelihood method. We have also applied this method to Monte Carlo simulations of the COBE -DMR data, as well as to the actual 4-yr data, obtaining consistent results with previous analyses. We also provide a very good analytical approximation to the distribution function of our statistic, which could also be useful in other contexts.     

Smoothing supernova data to reconstruct the expansion history of the Universe and its age

We propose a non-parametric method of smoothing supernova data over redshift using a Gaussian kernel in order to reconstruct important cosmological quantities including   H ( z )  and   w ( z )  in a model-independent manner. This method is shown to be successful in discriminating between different models of dark energy when the quality of data is commensurate with that expected from the future Supernova Acceleration Probe ( SNAP ). We find that the Hubble parameter is especially well determined and useful for this purpose. The look-back time of the Universe may also be determined to a very high degree of accuracy (≲0.2 per cent) using this method. By refining the method, it is also possible to obtain reasonable bounds on the equation of state of dark energy. We explore a new diagnostic of dark energy – the ' w -probe'– which can be calculated from the first derivative of the data. We find that this diagnostic is reconstructed extremely accurately for different reconstruction methods even if Ω_{0 m} is marginalized over. The w -probe can be used to successfully distinguish between Λ cold dark matter and other models of dark energy to a high degree of accuracy.     

On model selection forecasting, dark energy and modified gravity

The Fisher matrix approach allows one to calculate in advance how well a given experiment will be able to estimate model parameters, and has been an invaluable tool in experimental design. In the same spirit, we present here a method to predict how well a given experiment can distinguish between different models, regardless of their parameters. From a Bayesian viewpoint, this involves computation of the Bayesian evidence. In this paper, we generalize the Fisher matrix approach from the context of parameter fitting to that of model testing, and show how the expected evidence can be computed under the same simplifying assumption of a Gaussian likelihood as the Fisher matrix approach for parameter estimation. With this 'Laplace approximation' all that is needed to compute the expected evidence is the Fisher matrix itself. We illustrate the method with a study of how well upcoming and planned experiments should perform at distinguishing between dark energy models and modified gravity theories. In particular, we consider the combination of 3D weak lensing, for which planned and proposed wide-field multiband imaging surveys will provide suitable data, and probes of the expansion history of the Universe, such as proposed supernova and baryonic acoustic oscillations surveys. We find that proposed large-scale weak-lensing surveys from space should be able readily to distinguish General Relativity from modified gravity models.     

Monte‐Carlo simulation of a possible correlation between the estimated luminosity distances and internal extinctions of type Ia supernovae

This paper describes a Monte Carlo simulation of type Ia supernova data. It was shown earlier that the data of SNe Ia might contain a possible correlation between the estimated luminosity distances and internal extinctions. This correlation was shown by different statistical investigations of the data. In order to remove observational biases (for example the effect of the detection limit of the observing instrument) and to test the reality of the effect found earlier we developed a simple routine which simulates extinction values, redshifts and absolute magnitudes for Ia supernovae. We pointed out that the correlation found earlier in the real data between the internal extinction and luminosity distance does not occur in the simulated sample. Furthermore, it became obvious that the detection limit of the observing devices used in supernova projects does not affect the far end of the redshift‐luminosity distance relationship of Ia supernovae. This result strengthens the earlier conclusions of the authors that SN Ia supernovae alone do not support the existence of dark energy. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust method for measuring the Hubble parameter

We obtain a robust, non-parametric, estimate of the Hubble constant from the linear diameters and rotation velocities of galaxies in the recent KLUN sample, calibrated using Cepheid distances to Hubble Space Telescope Key Project galaxies. There are two key features that make our analysis considerably more robust than previous work. First, the method is independent of the spatial distribution of galaxies and is insensitive to Malmquist bias. It may, therefore, be applied to more distant samples than so-called 'plateau' methods – making it much less vulnerable to the impact of peculiar motions in the Local Supercluster. Secondly, we include information on the galaxy rotation velocities in a fully non-parametric manner: unlike the conventional Tully–Fisher relation we reconstruct a robust estimate of the cumulative distribution function of galaxy diameter at given rotation velocity, without requiring the assumption of, for example, a linear Tully–Fisher relation with symmetric Gaussian residuals.
Using this robust method we find H ₀=65±6 km s⁻¹ Mpc⁻¹ from our analysis – in excellent agreement with many recent determinations of the Hubble parameter, although somewhat larger than previous results using galaxy diameters.