Orthogonalized smoothing for rescaled spike and slab models

Rescaled spike and slab models are a new Bayesian variable selection method for linear regression models. In high dimensional orthogonal settings such models have been shown to possess optimal model selection properties. We review background theory and discuss applications of rescaled spike and slab models to prediction problems involving orthogonal polynomials. We first consider global smoothing and discuss potential weaknesses. Some of these deficiencies are remedied by using local regression. The local regression approach relies on an intimate connection between local weighted regression and weighted generalized ridge regression. An important implication is that one can trace the effective degrees of freedom of a curve as a way to visualize and classify curvature. Several motivating examples are presented.Comment: Published in at http://dx.doi.org/10.1214/074921708000000192 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The expansion rate of the intermediate Universe in light of Planck

We use cosmology-independent measurements of the expansion history in the redshift range 0.1 < z <1.2 and compare them with the Cosmic Microwave Background-derived expansion history predictions. The motivation is to investigate if the tension between the local (cosmology independent) Hubble constant H0 value and the Planck-derived H0 is also present at other redshifts. We conclude that there is no tension between Planck and cosmology independent-measurements of the Hubble parameter H(z) at 0.1 < z < 1.2 for the LCDM model (odds of tension are only 1:15, statistically not significant). Considering extensions of the LCDM model does not improve these odds (actually makes them worse), thus favouring the simpler model over its extensions. On the other hand the H(z) data are also not in tension with the local H0 measurements but the combination of all three data-sets shows a highly significant tension (odds ~ 1:400). Thus the new data deepen the mystery of the mismatch between Planck and local H0 measurements, and cannot univocally determine wether it is an effect localised at a particular redshift. Having said this, we find that assuming the NGC4258 maser distance as the correct anchor for H0, brings the odds to comfortable values. Further, using only the expansion history measurements we constrain, within the LCDM model, H0 = 68.5 +- 3.5 and Omega_m = 0.32 +- 0.05 without relying on any CMB prior. We also address the question of how smooth the expansion history of the universe is given the cosmology independent data and conclude that there is no evidence for deviations from smoothness on the expansion history, neither variations with time in the value of the equation of state of dark energy.Comment: Submitted to Physics of the Dark Univers

A Bayesian palaeoenvironmental transfer function model for acidified lakes

A Bayesian approach to palaeoecological environmental reconstruction deriving from the unimodal responses generally exhibited by organisms to an environmental gradient is described. The approach uses Bayesian model selection to calculate a collection of probability-weighted, species-specific response curves (SRCs) for each taxon within a training set, with an explicit treatment for zero abundances. These SRCs are used to reconstruct the environmental variable from sub-fossilised assemblages. The approach enables a substantial increase in computational efficiency (several orders of magnitude) over existing Bayesian methodologies. The model is developed from the Surface Water Acidification Programme (SWAP) training set and is demonstrated to exhibit comparable predictive power to existing Weighted Averaging and Maximum Likelihood methodologies, though with improvements in bias; the additional explanatory power of the Bayesian approach lies in an explicit calculation of uncertainty for each individual reconstruction. The model is applied to reconstruct the Holocene acidification history of the Round Loch of Glenhead, including a reconstruction of recent recovery derived from sediment trap data.The Bayesian reconstructions display similar trends to conventional (Weighted Averaging Partial Least Squares) reconstructions but provide a better reconstruction of extreme pH and are more sensitive to small changes in diatom assemblages. The validity of the posteriors as an apparently meaningful representation of assemblage-specific uncertainty and the high computational efficiency of the approach open up the possibility of highly constrained multiproxy reconstructions

Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio

We provide a new interpretation for the Bayes factor combination used in the Dark Energy Survey (DES) first year analysis to quantify the tension between the DES and Planck datasets. The ratio quantifies a Bayesian confidence in our ability to combine the datasets. This interpretation is prior-dependent, with wider prior widths boosting the confidence. We therefore propose that if there are any reasonable priors which reduce the confidence to below unity, then we cannot assert that the datasets are compatible. Computing the evidence ratios for the DES first year analysis and Planck, given that narrower priors drop the confidence to below unity, we conclude that DES and Planck are, in a Bayesian sense, incompatible under LCDM. Additionally we compute ratios which confirm the consensus that measurements of the acoustic scale by the Baryon Oscillation Spectroscopic Survey (SDSS) are compatible with Planck, whilst direct measurements of the acceleration rate of the Universe by the SHOES collaboration are not. We propose a modification to the Bayes ratio which removes the prior dependency using Kullback-Leibler divergences, and using this statistical test find Planck in strong tension with SHOES, in moderate tension with DES, and in no tension with SDSS. We propose this statistic as the optimal way to compare datasets, ahead of the next DES data releases, as well as future surveys. Finally, as an element of these calculations, we introduce in a cosmological setting the Bayesian model dimensionality, which is a parameterisation-independent measure of the number of parameters that a given dataset constrains.Comment: 16 pages, 9 figures. v2 & v3: updates post peer-review. v4: typographical correction to the reported errors in the log S column of Table II. v5: typographical correction to equation 2

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page