Semiparametric posterior limits

We review the Bayesian theory of semiparametric inference following Bickel and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency in parametric and semiparametric estimation problems, we consider the Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We formulate a version of the semiparametric Bernstein-von Mises theorem that does not depend on least-favourable submodels, thus bypassing the most restrictive condition in the presentation of Bickel and Kleijn (2012). The results are applied to the (regular) estimation of the linear coefficient in partial linear regression (with a Gaussian nuisance prior) and of the kernel bandwidth in a model of normal location mixtures (with a Dirichlet nuisance prior), as well as the (irregular) estimation of the boundary of the support of a monotone family of densities (with a Gaussian nuisance prior).Comment: 47 pp., 1 figure, submitted for publication. arXiv admin note: substantial text overlap with arXiv:1007.017

Recovery, detection and confidence sets of communities in a sparse stochastic block model

Posterior distributions for community assignment in the planted bi-section model are shown to achieve frequentist exact recovery and detection under sharp lower bounds on sparsity. Assuming posterior recovery (or detection), one may interpret credible sets (or enlarged credible sets) as consistent confidence sets. If credible levels grow to one quickly enough, credible sets can be interpreted as frequentist confidence sets without conditions on the parameters. In the regime where within-class and between-class edge-probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage. The diameters of credible sets are controlled and match rates of posterior convergence.Comment: 22 pp., 2 fi

The semiparametric Bernstein-von Mises theorem

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam's acclaimed, but strictly parametric, Bernstein-von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277-329] holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example, in the sense of H\'{a}jek's convolution theorem [Z. Wahrsch. Verw. Gebiete 14 (1970) 323-330]. The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign nonzero mass to certain Kullback-Leibler neighborhoods [Ghosal, Ghosh and van der Vaart Ann. Statist. 28 (2000) 500-531]. In addition, the marginal posterior is required to converge at parametric rate, which appears to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior on a smoothness class for the nuisance.Comment: Published in at http://dx.doi.org/10.1214/11-AOS921 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Bayesian Analysis of Complex, High-Dimensional Models: Can It Be CODA?

We consider the Bayesian analysis of a few complex, high-dimensional models and show that intuitive priors, which are not tailored to the fine details of the model and the estimated parameters, produce estimators which perform poorly in situations in which good, simple frequentist estimators exist. The models we consider are: stratified sampling, the partial linear model, linear and quadratic functionals of white noise and estimation with stopping times. We present a strong version of Doob's consistency theorem which demonstrates that the existence of a uniformly $\sqrt{n}$-consistent estimator ensures that the Bayes posterior is $\sqrt{n}$-consistent for values of the parameter in subsets of prior probability 1. We also demonstrate that it is, at least, in principle, possible to construct Bayes priors giving both global and local minimax rates, using a suitable combination of loss functions. We argue that there is no contradiction in these apparently conflicting findings.Comment: Published in at http://dx.doi.org/10.1214/14-STS483 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors

We consider the horseshoe estimator due to Carvalho, Polson and Scott (2010) for the multivariate normal mean model in the situation that the mean vector is sparse in the nearly black sense. We assume the frequentist framework where the data is generated according to a fixed mean vector. We show that if the number of nonzero parameters of the mean vector is known, the horseshoe estimator attains the minimax $\ell_2$ risk, possibly up to a multiplicative constant. We provide conditions under which the horseshoe estimator combined with an empirical Bayes estimate of the number of nonzero means still yields the minimax risk. We furthermore prove an upper bound on the rate of contraction of the posterior distribution around the horseshoe estimator, and a lower bound on the posterior variance. These bounds indicate that the posterior distribution of the horseshoe prior may be more informative than that of other one-component priors, including the Lasso.Comment: This version differs from the final published version in pagination and typographical detail; Available at http://projecteuclid.org/euclid.ejs/141813426

Target and (Astro-)WISE technologies - Data federations and its applications

After its first implementation in 2003 the Astro-WISE technology has been rolled out in several European countries and is used for the production of the KiDS survey data. In the multi-disciplinary Target initiative this technology, nicknamed WISE technology, has been further applied to a large number of projects. Here, we highlight the data handling of other astronomical applications, such as VLT-MUSE and LOFAR, together with some non-astronomical applications such as the medical projects Lifelines and GLIMPS, the MONK handwritten text recognition system, and business applications, by amongst others, the Target Holding. We describe some of the most important lessons learned and describe the application of the data-centric WISE type of approach to the Science Ground Segment of the Euclid satellite.Comment: 9 pages, 5 figures, Proceedngs IAU Symposium No 325 Astroinformatics 201

Star formation in z>1 3CR host galaxies as seen by Herschel

We present Herschel (PACS and SPIRE) far-infrared (FIR) photometry of a complete sample of z>1 3CR sources, from the Herschel GT project The Herschel Legacy of distant radio-loud AGN (PI: Barthel). Combining these with existing Spitzer photometric data, we perform an infrared (IR) spectral energy distribution (SED) analysis of these landmark objects in extragalactic research to study the star formation in the hosts of some of the brightest active galactic nuclei (AGN) known at any epoch. Accounting for the contribution from an AGN-powered warm dust component to the IR SED, about 40% of our objects undergo episodes of prodigious, ULIRG-strength star formation, with rates of hundreds of solar masses per year, coeval with the growth of the central supermassive black hole. Median SEDs imply that the quasar and radio galaxy hosts have similar FIR properties, in agreement with the orientation-based unification for radio-loud AGN. The star-forming properties of the AGN hosts are similar to those of the general population of equally massive non-AGN galaxies at comparable redshifts, thus there is no strong evidence of universal quenching of star formation (negative feedback) within this sample. Massive galaxies at high redshift may be forming stars prodigiously, regardless of whether their supermassive black holes are accreting or not.Comment: 30 pages, 13 figures, 4 tables. Accepted for publication in A&

Measuring the mass of the central black hole in the bulgeless galaxy ngc 4395 from gas dynamical modeling

NGC 4395 is a bulgeless spiral galaxy, harboring one of the nearest known type 1 Seyfert nuclei. Although there is no consensus on the mass of its central engine, several estimates suggest it is one of the lightest massive black holes (MBHs) known. We present the first direct dynamical measurement of the mass of this MBH from a combination of two-dimensional gas kinematic data, obtained with the adaptive optics assisted near-infrared integral field spectrograph Gemini/NIFS and high-resolution multiband photometric data from Hubble Space Telescope's Wide Field Camera 3. We use the photometric data to model the shape and stellar mass-to-light ratio of the nuclear star cluster (NSC). From the Gemini/NIFS observations, we derive the kinematics of warm molecular hydrogen gas as traced by emission through the H2 1–0 S(1) transition. These kinematics show a clear rotational signal, with a position angle orthogonal to NGC 4395's radio jet. Our best-fitting tilted ring models of the kinematics of the molecular hydrogen gas contain a black hole with mass M={4}-3+8× {10}5 M⊙ (3σ uncertainties) embedded in an NSC of mass M=2× {10}6 M⊙. Our black hole mass measurement is in excellent agreement with the reverberation mapping mass estimate of Peterson et al. but shows some tension with other mass measurement methods based on accretion signals