Rates of contraction of posterior distributions based on Gaussian process priors

We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion. We also consider regression with Gaussian errors and smooth classification under a logistic or probit link function combined with various priors.Comment: Published in at http://dx.doi.org/10.1214/009053607000000613 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth

We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax-optimal up to a logarithmic factor, whatever the regularity level of the data-generating distribution. Thus the hierachical Bayesian procedure, with a fixed prior, is shown to be fully adaptive.Comment: Published in at http://dx.doi.org/10.1214/08-AOS678 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Frequentist coverage of adaptive nonparametric Bayesian credible sets

We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of "polished tail" parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale.Comment: Published at http://dx.doi.org/10.1214/14-AOS1270 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rejoinder to discussions of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"

Rejoinder of "Frequentist coverage of adaptive nonparametric Bayesian credible sets" by Szab\'o, van der Vaart and van Zanten [arXiv:1310.4489v5].Comment: Published at http://dx.doi.org/10.1214/15-AOS1270REJ in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian inverse problems with Gaussian priors

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage. In the latter case credible sets are of the correct order of magnitude. The results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive.Comment: Published in at http://dx.doi.org/10.1214/11-AOS920 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mapping Landscape Values Using Social Media

Social media data are providing scientists with a variety of new ways to examine how and why individuals value particular natural landscapes. In this fact sheet, we review cutting edge research that used millions of photos posted to Instagram, Flickr and Panaramio to examine which European landscapes individuals value most. The research is the first of its kind to use social media data to identify the public’s most valued landscapes across an entire continent. The research is also the first to compare the spatial agreement between geotagged imagery uploaded to different platforms

Tapping into Social Media Data to Identify the Public\u27s Most Valued Landscapes

Today, millions of people are using social media to share information and images about the places they visit for outdoor recreation and leisure. This fact sheet reviews recent research which analyzed over 7.5 million photos posted to Instagram, Flickr, and Panaramio to examine which European landscapes individuals value most. The research is the first of its kind to use social media data to identify the public’s most valued landscapes across an entire continent

The effect of cisatracurium infusion on the energy expenditure of critically ill patients: An observational cohort study

Background: Both overfeeding and underfeeding of intensive care unit (ICU) patients are associated with worse outcomes. A reliable estimation of the energy expenditure (EE) of ICU patients may help to avoid these phenomena. Several factors that influence EE have been studied previously. However, the effect of neuromuscular blocking agents on EE, which conceptually would lower EE, has not been extensively investigated. Methods: We studied a cohort of adult critically ill patients requiring invasive mechanical ventilation and treatment with continuous infusion of cisatracurium for at least 12 h. The study aimed to quantify the effect of cisatracurium infusion on EE (primary endpoint). EE was estimated based on ventilator-derived VCO2 (EE in kcal/day = VCO2 × 8.19). A subgroup analysis of septic and non-septic patients was performed. Furthermore, the effects of body temperature and sepsis on EE were evaluated. A secondary endpoint was hypercaloric feeding (> 110% of EE) after cisatracurium infusion. Results: In total, 122 patients were included. Mean EE before cisatracurium infusion was 1974 kcal/day and 1888 kcal/day after cisatracurium infusion. Multivariable analysis showed a significantly lower EE after cisatracurium infusion (MD - 132.0 kcal (95% CI - 212.0 to - 52.0; p = 0.001) in all patients. This difference was statistically significant in both sepsis and non-sepsis patients (p = 0.036 and p = 0.011). Non-sepsis patients had lower EE than sepsis patients (MD - 120.6 kcal; 95% CI - 200.5 to - 40.8, p = 0.003). Body temperature and EE were positively correlated (Spearman's rho = 0.486, p < 0.001). Hypercaloric feeding was observed in 7 patients. Conclusions: Our data suggest that continuous infusion of cisatracurium in mechanically ventilated ICU patients is associated with a significant reduction in EE, although the magnitude of the effect is small. Sepsis and higher body temperature are associated with increased EE. Cisatracurium infusion is associated with overfeeding in only a minority of patients and therefore, in most patients, no reductions in caloric prescription are necessary

Bayesian recovery of the initial condition for the heat equation

We study a Bayesian approach to recovering the initial condition for the heat equation from noisy observations of the solution at a later time. We consider a class of prior distributions indexed by a parameter quantifying "smoothness" and show that the corresponding posterior distributions contract around the true parameter at a rate that depends on the smoothness of the true initial condition and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the optimal minimax rate. One type of priors leads to a rate-adaptive Bayesian procedure. The frequentist coverage of credible sets is shown to depend on the combination of the prior and true parameter as well, with smoother priors leading to zero coverage and rougher priors to (extremely) conservative results. In the latter case credible sets are much larger than frequentist confidence sets, in that the ratio of diameters diverges to infinity. The results are numerically illustrated by a simulated data example.Comment: 17 pages, 4 figures. Published in Comm. Statist. Theory Methods. This version differs from the original in pagination and typographic detail. arXiv admin note: text overlap with arXiv:1103.269