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Author: Giuseppe De Luca
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Languages : en
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Book Description
Many statistical and econometric learning methods rely on Bayesian ideas, often applied or reinterpreted in a frequentist setting. Two leading examples are shrinkage estimators and model averaging estimators, such as weighted-average least squares (WALS). In many instances, the accuracy of these learning methods in repeated samples is assessed using the variance of the posterior distribution of the parameters of interest given the data. This may be permissible when the sample size is large because, under the conditions of the Bernstein-von Mises theorem, the posterior variance agrees asymptotically with the frequentist variance. In finite samples, however, things are less clear. In this paper we explore this issue by first considering the frequentist properties (bias and variance) of the posterior mean in the important case of the normal location model, which consists of a single observation on a univariate Gaussian distribution with unknown mean and known variance. Based on these results, we derive new estimators of the frequentist bias and variance of the WALS estimator in finite samples. We then study the finite-sample performance of the proposed estimators by a Monte Carlo experiment with design derived from a real data application about the effect of abortion on crime rates.
Author: Giuseppe De Luca
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ISBN:
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Languages : en
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Book Description
Many statistical and econometric learning methods rely on Bayesian ideas, often applied or reinterpreted in a frequentist setting. Two leading examples are shrinkage estimators and model averaging estimators, such as weighted-average least squares (WALS). In many instances, the accuracy of these learning methods in repeated samples is assessed using the variance of the posterior distribution of the parameters of interest given the data. This may be permissible when the sample size is large because, under the conditions of the Bernstein-von Mises theorem, the posterior variance agrees asymptotically with the frequentist variance. In finite samples, however, things are less clear. In this paper we explore this issue by first considering the frequentist properties (bias and variance) of the posterior mean in the important case of the normal location model, which consists of a single observation on a univariate Gaussian distribution with unknown mean and known variance. Based on these results, we derive new estimators of the frequentist bias and variance of the WALS estimator in finite samples. We then study the finite-sample performance of the proposed estimators by a Monte Carlo experiment with design derived from a real data application about the effect of abortion on crime rates.
Author: Giuseppe De Luca
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ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: Giuseppe De Luca
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
We investigate the asymptotic behavior of the WALS estimator, a model-averaging estimator with attractive finite-sample and computational properties. WALS is closely related to the normal location model, and hence much of the paper concerns the asymptotic behavior of the estimator of the unknown mean in the normal local model. Since we adopt a frequentist-Bayesian approach, this specializes to the asymptotic behavior of the posterior mean as a frequentist estimator of the normal location parameter. We emphasize two challenging issues. First, our definition of ignorance in the Bayesian step involves a prior on the t-ratio rather than on the parameter itself. Second, instead of assuming a local misspecification framework, we consider a standard asymptotic setup with fixed parameters. We show that, under suitable conditions on the prior, the WALS estimator is √n-consistent and its asymptotic distribution essentially coincides with that of the unrestricted least-squares estimator. Monte Carlo simulations confirm our theoretical results.
Author: Paul Gustafson
Publisher: CRC Press
ISBN: 1439869405
Category : Mathematics
Languages : en
Pages : 196
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Book Description
Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data shows how the Bayesian approach to inference is applicable to partially identified models (PIMs) and examines the performance of Bayesian procedures in partially identified contexts. Drawing on his many years of research in this area, the author presents a thorough overview of the statistical theory, properties, and applications of PIMs. The book first describes how reparameterization can assist in computing posterior quantities and providing insight into the properties of Bayesian estimators. It next compares partial identification and model misspecification, discussing which is the lesser of the two evils. The author then works through PIM examples in depth, examining the ramifications of partial identification in terms of how inferences change and the extent to which they sharpen as more data accumulate. He also explains how to characterize the value of information obtained from data in a partially identified context and explores some recent applications of PIMs. In the final chapter, the author shares his thoughts on the past and present state of research on partial identification. This book helps readers understand how to use Bayesian methods for analyzing PIMs. Readers will recognize under what circumstances a posterior distribution on a target parameter will be usefully narrow versus uselessly wide.
Author: Malay Ghosh
Publisher: CRC Press
ISBN: 9780412987717
Category : Mathematics
Languages : en
Pages : 304
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Book Description
Assuming a basic knowledge of the frequentist approach to finite population sampling, Bayesian Methods for Finite Population Sampling describes Bayesian and predictive approaches to inferential problems with an emphasis on the likelihood principle. The authors demonstrate that a variety of levels of prior information can be used in survey sampling in a Bayesian manner. Situations considered range from a noninformative Bayesian justification of standard frequentist methods when the only prior information available is the belief in the exchangeability of the units to a full-fledged Bayesian model. Intended primarily for graduate students and researchers in finite population sampling, this book will also be of interest to statisticians who use sampling and lecturers and researchers in general statistics and biostatistics.
Author: Lawrence D. Phillips
Publisher:
ISBN:
Category : Bayesian statistical decision theory
Languages : en
Pages : 472
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Book Description
Author: Brian J. Reich
Publisher: CRC Press
ISBN: 0429514344
Category : Mathematics
Languages : en
Pages : 197
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Book Description
Bayesian Statistical Methods provides data scientists with the foundational and computational tools needed to carry out a Bayesian analysis. This book focuses on Bayesian methods applied routinely in practice including multiple linear regression, mixed effects models and generalized linear models (GLM). The authors include many examples with complete R code and comparisons with analogous frequentist procedures. In addition to the basic concepts of Bayesian inferential methods, the book covers many general topics: Advice on selecting prior distributions Computational methods including Markov chain Monte Carlo (MCMC) Model-comparison and goodness-of-fit measures, including sensitivity to priors Frequentist properties of Bayesian methods Case studies covering advanced topics illustrate the flexibility of the Bayesian approach: Semiparametric regression Handling of missing data using predictive distributions Priors for high-dimensional regression models Computational techniques for large datasets Spatial data analysis The advanced topics are presented with sufficient conceptual depth that the reader will be able to carry out such analysis and argue the relative merits of Bayesian and classical methods. A repository of R code, motivating data sets, and complete data analyses are available on the book’s website. Brian J. Reich, Associate Professor of Statistics at North Carolina State University, is currently the editor-in-chief of the Journal of Agricultural, Biological, and Environmental Statistics and was awarded the LeRoy & Elva Martin Teaching Award. Sujit K. Ghosh, Professor of Statistics at North Carolina State University, has over 22 years of research and teaching experience in conducting Bayesian analyses, received the Cavell Brownie mentoring award, and served as the Deputy Director at the Statistical and Applied Mathematical Sciences Institute.
Author: Katherine Rose St. Clair
Publisher:
ISBN:
Category :
Languages : en
Pages : 346
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Book Description
Author: Borek Puza
Publisher: ANU Press
ISBN: 1921934263
Category : Mathematics
Languages : en
Pages : 698
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Book Description
Bayesian Methods for Statistical Analysis is a book on statistical methods for analysing a wide variety of data. The book consists of 12 chapters, starting with basic concepts and covering numerous topics, including Bayesian estimation, decision theory, prediction, hypothesis testing, hierarchical models, Markov chain Monte Carlo methods, finite population inference, biased sampling and nonignorable nonresponse. The book contains many exercises, all with worked solutions, including complete computer code. It is suitable for self-study or a semester-long course, with three hours of lectures and one tutorial per week for 13 weeks.
Author: Thomas Leonard
Publisher: Cambridge University Press
ISBN: 9780521004145
Category : Mathematics
Languages : en
Pages : 352
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Book Description
Bayesian statistics directed towards mainstream statistics. How to infer scientific, medical, and social conclusions from numerical data.
