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BERNSTEIN-VON MISES THEOREMS FOR GAUSSIAN REGRESSION WITH INCREASING NUMBER OF REGRESSORS
This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernsteinvon Mises theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and C α classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications.
Journal Article
Stationary stochastic processes : theory and applications
In recent years, applications of advanced stochastic processes have expanded greatly. Intended for students taking a second course in stochastic processes, this textbook presents an overview of theory with applications in engineering and science. This book covers key topics such as ergodicity, crossing problems, and extremes, and opens the doors to a selection of special topics for the teacher to expand on, like extreme value theory, filter theory, long-range dependence, and point processes, and includes many exercises and examples to illustrate the theory.
eBook
Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2)
Estimation of the prevalence and contagiousness of undocumented novel coronavirus [severe acute respiratory syndrome–coronavirus 2 (SARS-CoV-2)] infections is critical for understanding the overall prevalence and pandemic potential of this disease. Here, we use observations of reported infection within China, in conjunction with mobility data, a networked dynamic metapopulation model, and Bayesian inference, to infer critical epidemiological characteristics associated with SARS-CoV-2, including the fraction of undocumented infections and their contagiousness. We estimate that 86% of all infections were undocumented [95% credible interval (CI): 82–90%] before the 23 January 2020 travel restrictions. The transmission rate of undocumented infections per person was 55% the transmission rate of documented infections (95% CI: 46–62%), yet, because of their greater numbers, undocumented infections were the source of 79% of the documented cases. These findings explain the rapid geographic spread of SARS-CoV-2 and indicate that containment of this virus will be particularly challenging.
Journal Article
Bayesian analysis of stochastic process models
Bayesian analysis of complex models based on stochastic processes has in recent years become a growing area. This book provides a unified treatment of Bayesian analysis of models based on stochastic processes, covering the main classes of stochastic processing including modeling, computational, inference, forecasting, decision making and important applied models. Key features: * Explores Bayesian analysis of models based on stochastic processes, providing a unified treatment. * Provides a thorough introduction for research students. * Computational tools to deal with complex problems are illustrated along with real life case studies * Looks at inference, prediction and decision making. Researchers, graduate and advanced undergraduate students interested in stochastic processes in fields such as statistics, operations research (OR), engineering, finance, economics, computer science and Bayesian analysis will benefit from reading this book. With numerous applications included, practitioners of OR, stochastic modelling and applied statistics will also find this book useful.
eBook
Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array
Pulsar timing arrays aim to detect nanohertz-frequency gravitational waves (GWs). A background of GWs modulates pulsar arrival times and manifests as a stochastic process, common to all pulsars, with a signature spatial correlation. Here we describe a search for an isotropic stochastic gravitational-wave background (GWB) using observations of 30 millisecond pulsars from the third data release of the Parkes Pulsar Timing Array (PPTA), which spans 18 yr. Using current Bayesian inference techniques we recover and characterize a common-spectrum noise process. Represented as a strain spectrum hc=A(f/1yr−1)α , we measure A=3.1−0.9+1.3×10−15 and α = −0.45 ± 0.20, respectively (median and 68% credible interval). For a spectral index of α = −2/3, corresponding to an isotropic background of GWs radiated by inspiraling supermassive black hole binaries, we recover an amplitude of A=2.04−0.22+0.25×10−15 . However, we demonstrate that the apparent signal strength is time-dependent, as the first half of our data set can be used to place an upper limit on A that is in tension with the inferred common-spectrum amplitude using the complete data set. We search for spatial correlations in the observations by hierarchically analyzing individual pulsar pairs, which also allows for significance validation through randomizing pulsar positions on the sky. For a process with α = −2/3, we measure spatial correlations consistent with a GWB, with an estimated false-alarm probability of p ≲ 0.02 (approx. 2σ). The long timing baselines of the PPTA and the access to southern pulsars will continue to play an important role in the International Pulsar Timing Array.
Journal Article