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Bayesian Estimation of Intensity Surfaces on the Sphere via Needlet Shrinkage and Selection
This paper describes an approach for Bayesian modeling in spherical datasets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding, and Bayesian local shrinkage rules. We study the performance of the proposed methodology both on simulated data and on two real data sets: one involving the cosmic microwave background radiation, and one involving the reconstruction of a global news intensity surface inferred from published Reuters articles in August, 1996. The fully Bayesian approach based on robust, sparse shrinkage priors seems to outperform other alternatives.Business Administratio
Localizing merging black holes with sub-arcsecond precision using gravitational-wave lensing
The current gravitational-wave localization methods rely mainly on sources
with electromagnetic counterparts. Unfortunately, a binary black hole does not
emit light. Due to this, it is generally not possible to localize these objects
precisely. However, strongly lensed gravitational waves, which are forecasted
in this decade, could allow us to localize the binary by locating its lensed
host galaxy. Identifying the correct host galaxy is challenging because there
are hundreds to thousands of other lensed galaxies within the sky area spanned
by the gravitational-wave observation. However, we can constrain the lensing
galaxy's physical properties through both gravitational-wave and
electromagnetic observations. We show that these simultaneous constraints allow
one to localize quadruply lensed waves to one or at most a few galaxies with
the LIGO/Virgo/Kagra network in typical scenarios. Once we identify the host,
we can localize the binary to two sub-arcsec regions within the host galaxy.
Moreover, we demonstrate how to use the system to measure the Hubble constant
as a proof-of-principle application.Comment: 5 pages (main text) + 5 pages (methods+references), 5 figures.
Accepted to MNRA
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
CARBayes: an R package for Bayesian spatial modeling with conditional autoregressive priors
Conditional autoregressive models are commonly used to represent spatial autocorrelation in data relating to a set of non-overlapping areal units, which arise in a wide variety of applications including agriculture, education, epidemiology and image analysis. Such models are typically specified in a hierarchical Bayesian framework, with inference based on Markov chain Monte Carlo (MCMC) simulation. The most widely used software to fit such models is WinBUGS or OpenBUGS, but in this paper we introduce the R package CARBayes. The main advantage of CARBayes compared with the BUGS software is its ease of use, because: (1) the spatial adjacency information is easy to specify as a binary neighbourhood matrix; and (2) given the neighbourhood matrix the models can be implemented by a single function call in R. This paper outlines the general class of Bayesian hierarchical models that can be implemented in the CARBayes software, describes their implementation via MCMC simulation techniques, and illustrates their use with two worked examples in the fields of house price analysis and disease mapping
Fundamental remote sensing science research program. Part 1: Status report of the mathematical pattern recognition and image analysis project
The Mathematical Pattern Recognition and Image Analysis (MPRIA) Project is concerned with basic research problems related to the study of the Earth from remotely sensed measurement of its surface characteristics. The program goal is to better understand how to analyze the digital image that represents the spatial, spectral, and temporal arrangement of these measurements for purposing of making selected inference about the Earth
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
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