To shed light on the fundamental problems posed by dark energy and dark matter, a large number of experiments have been performed and combined to constrain cosmological models. We propose a novel way of quantifying the information gained by updates on the parameter constraints from a series of experiments which can either complement earlier measurements or replace them. For this purpose, we use the Kullback-Leibler divergence or relative entropy from information theory to measure differences in the posterior distributions in model parameter space from a pair of experiments. We apply this formalism to a historical series of cosmic microwave background experiments ranging from Boomerang to WMAP, SPT, and Planck. Considering different combinations of these experiments, we thus estimate the information gain in units of bits and distinguish contributions from the reduction of statistical errors and the "surprise" corresponding to a significant shift of the parameters' central values. For this experiment series, we find individual relative entropy gains ranging from about 1 to 30 bits. In some cases, e.g. when comparing WMAP and Planck results, we find that the gains are dominated by the surprise rather than by improvements in statistical precision. We discuss how this technique provides a useful tool for both quantifying the constraining power of data from cosmological probes and detecting the tensions between experiments.
Abstract. Bayesian inference is often used in cosmology and astrophysics to derive constraints on model parameters from observations. This approach relies on the ability to compute the likelihood of the data given a choice of model parameters. In many practical situations, the likelihood function may however be unavailable or intractable due to non-gaussian errors, non-linear measurements processes, or complex data formats such as catalogs and maps. In these cases, the simulation of mock data sets can often be made through forward modeling. We discuss how Approximate Bayesian Computation (ABC) can be used in these cases to derive an approximation to the posterior constraints using simulated data sets. This technique relies on the sampling of the parameter set, a distance metric to quantify the difference between the observation and the simulations and summary statistics to compress the information in the data. We first review the principles of ABC and discuss its implementation using a Population Monte-Carlo (PMC) algorithm and the Mahalanobis distance metric. We test the performance of the implementation using a Gaussian toy model. We then apply the ABC technique to the practical case of the calibration of image simulations for wide field cosmological surveys. We find that the ABC analysis is able to provide reliable parameter constraints for this problem and is therefore a promising technique for other applications in cosmology and astrophysics. Our implementation of the ABC PMC method is made available via a public code release.
We study the benefits and limits of parallelised Markov chain Monte Carlo (MCMC) sampling in cosmology. MCMC methods are widely used for the estimation of cosmological parameters from a given set of observations and are typically based on the Metropolis-Hastings algorithm. Some of the required calculations can however be computationally intensive, meaning that a single long chain can take several hours or days to calculate. In practice, this can be limiting, since the MCMC process needs to be performed many times to test the impact of possible systematics and to understand the robustness of the measurements being made. To achieve greater speed through parallelisation, MCMC algorithms need to have short auto-correlation times and minimal overheads caused by tuning and burn-in. The resulting scalability is hence influenced by two factors, the MCMC overheads and the parallelisation costs. In order to efficiently distribute the MCMC sampling over thousands of cores on modern cloud computing infrastructure, we developed a Python framework called CosmoHammer which embeds emcee, an implementation by Foreman-Mackey et al. (2012) of the affine invariant ensemble sampler by Goodman and Weare (2010). We test the performance of CosmoHammer for cosmological parameter estimation from cosmic microwave background data. While Metropolis-Hastings is dominated by overheads, CosmoHammer is able to accelerate the sampling process from a wall time of 30 hours on a dual core notebook to 16 minutes by scaling out to 2048 cores. Such short wall times for complex data sets opens possibilities for extensive model testing and control of systematics.Comment: Published version. 17 pages, 6 figures. The code is available at http://www.astro.ethz.ch/refregier/research/Software/cosmohamme
Quantifying the concordance between different cosmological experiments is important for testing the validity of theoretical models and systematics in the observations. In earlier work, we thus proposed the Surprise, a concordance measure derived from the relative entropy between posterior distributions. We revisit the properties of the Surprise and describe how it provides a general, versatile, and robust measure for the agreement between datasets. We also compare it to other measures of concordance that have been proposed for cosmology. As an application, we extend our earlier analysis and use the Surprise to quantify the agreement between WMAP 9, Planck 13 and Planck 15 constraints on the ΛCDM model. Using a principle component analysis in parameter space, we find that the large Surprise between WMAP 9 and Planck 13 (S = 17.6 bits, implying a deviation from consistency at 99.8% confidence) is due to a shift along a direction that is dominated by the amplitude of the power spectrum. The Planck 15 constraints deviate from the Planck 13 results (S = 56.3 bits), primarily due to a shift in the same direction. The Surprise between WMAP and Planck consequently disappears when moving to Planck 15 (S = −5.1 bits). This means that, unlike Planck 13, Planck 15 is not in tension with WMAP 9. These results illustrate the advantages of the relative entropy and the Surprise for quantifying the disagreement between cosmological experiments and more generally as an information metric for cosmology.
Abstract. In light of the growing number of cosmological observations, it is important to develop versatile tools to quantify the constraining power and consistency of cosmological probes. Originally motivated from information theory, we use the relative entropy to compute the information gained by Bayesian updates in units of bits. This measure quantifies both the improvement in precision and the 'surprise', i.e. the tension arising from shifts in central values. Our starting point is a WMAP9 prior which we update with observations of the distance ladder, supernovae (SNe), baryon acoustic oscillations (BAO), and weak lensing as well as the 2015 Planck release. We consider the parameters of the flat ΛCDM concordance model and some of its extensions which include curvature and Dark Energy equation of state parameter w. We find that, relative to WMAP9 and within these model spaces, the probes that have provided the greatest gains are Planck (10 bits), followed by BAO surveys (5.1 bits) and SNe experiments (3.1 bits). The other cosmological probes, including weak lensing (1.7 bits) and H 0 measures (1.7 bits), have contributed information but at a lower level. Furthermore, we do not find any significant surprise when updating the constraints of WMAP9 with any of the other experiments, meaning that they are consistent with WMAP9. However, when we choose Planck15 as the prior, we find that, accounting for the full multidimensionality of the parameter space, the weak lensing measurements of CFHTLenS produce a large surprise of 4.4 bits which is statistically significant at the 8 σ level. We discuss how the relative entropy provides a versatile and robust framework to compare cosmological probes in the context of current and future surveys.
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