Likelihood and Bayesian Inference

Likelihood and Bayesian Inference

This revised edition of the book “Applied Statistical Inference” has been expanded to include new material on Markov models for time series analysis.

Author: Leonhard Held

Publisher: Springer

ISBN: 3662607913

Category: Medical

Page: 402

View: 226

This richly illustrated textbook covers modern statistical methods with applications in medicine, epidemiology and biology. Firstly, it discusses the importance of statistical models in applied quantitative research and the central role of the likelihood function, describing likelihood-based inference from a frequentist viewpoint, and exploring the properties of the maximum likelihood estimate, the score function, the likelihood ratio and the Wald statistic. In the second part of the book, likelihood is combined with prior information to perform Bayesian inference. Topics include Bayesian updating, conjugate and reference priors, Bayesian point and interval estimates, Bayesian asymptotics and empirical Bayes methods. It includes a separate chapter on modern numerical techniques for Bayesian inference, and also addresses advanced topics, such as model choice and prediction from frequentist and Bayesian perspectives. This revised edition of the book “Applied Statistical Inference” has been expanded to include new material on Markov models for time series analysis. It also features a comprehensive appendix covering the prerequisites in probability theory, matrix algebra, mathematical calculus, and numerical analysis, and each chapter is complemented by exercises. The text is primarily intended for graduate statistics and biostatistics students with an interest in applications.
Categories: Medical

Applied Statistical Inference

Applied Statistical Inference

This book covers modern statistical inference based on likelihood with applications in medicine, epidemiology and biology.

Author: Leonhard Held

Publisher: Springer Science & Business Media

ISBN: 9783642378874

Category: Mathematics

Page: 376

View: 626

This book covers modern statistical inference based on likelihood with applications in medicine, epidemiology and biology. Two introductory chapters discuss the importance of statistical models in applied quantitative research and the central role of the likelihood function. The rest of the book is divided into three parts. The first describes likelihood-based inference from a frequentist viewpoint. Properties of the maximum likelihood estimate, the score function, the likelihood ratio and the Wald statistic are discussed in detail. In the second part, likelihood is combined with prior information to perform Bayesian inference. Topics include Bayesian updating, conjugate and reference priors, Bayesian point and interval estimates, Bayesian asymptotics and empirical Bayes methods. Modern numerical techniques for Bayesian inference are described in a separate chapter. Finally two more advanced topics, model choice and prediction, are discussed both from a frequentist and a Bayesian perspective. A comprehensive appendix covers the necessary prerequisites in probability theory, matrix algebra, mathematical calculus, and numerical analysis.
Categories: Mathematics

Empirical Bayes and Likelihood Inference

Empirical Bayes and Likelihood Inference

In this volume, researchers present recent work on several aspects of Bayesian, likelihood and empirical Bayes methods, presented at a workshop held in Montreal, Canada.

Author: S.E. Ahmed

Publisher: Springer Science & Business Media

ISBN: 9781461301417

Category: Mathematics

Page: 235

View: 876

Bayesian and such approaches to inference have a number of points of close contact, especially from an asymptotic point of view. Both emphasize the construction of interval estimates of unknown parameters. In this volume, researchers present recent work on several aspects of Bayesian, likelihood and empirical Bayes methods, presented at a workshop held in Montreal, Canada. The goal of the workshop was to explore the linkages among the methods, and to suggest new directions for research in the theory of inference.
Categories: Mathematics

Statistical Inference

Statistical Inference

This novel approach provides new solutions to difficult model comparison problems and offers direct

Author: Murray Aitkin

Publisher: CRC Press

ISBN: 9781420093445

Category: Mathematics

Page: 254

View: 397

Filling a gap in current Bayesian theory, Statistical Inference: An Integrated Bayesian/Likelihood Approach presents a unified Bayesian treatment of parameter inference and model comparisons that can be used with simple diffuse prior specifications. This novel approach provides new solutions to difficult model comparison problems and offers direct Bayesian counterparts of frequentist t-tests and other standard statistical methods for hypothesis testing. After an overview of the competing theories of statistical inference, the book introduces the Bayes/likelihood approach used throughout. It presents Bayesian versions of one- and two-sample t-tests, along with the corresponding normal variance tests. The author then thoroughly discusses the use of the multinomial model and noninformative Dirichlet priors in "model-free" or nonparametric Bayesian survey analysis, before covering normal regression and analysis of variance. In the chapter on binomial and multinomial data, he gives alternatives, based on Bayesian analyses, to current frequentist nonparametric methods. The text concludes with new goodness-of-fit methods for assessing parametric models and a discussion of two-level variance component models and finite mixtures. Emphasizing the principles of Bayesian inference and Bayesian model comparison, this book develops a unique methodology for solving challenging inference problems. It also includes a concise review of the various approaches to inference.
Categories: Mathematics

Likelihood and Bayesian Inference

Likelihood and Bayesian Inference

This book covers statistical inference based on the likelihood function.

Author: Leonhard Held

Publisher:

ISBN: 366260793X

Category: Bayesian statistical decision theory

Page: 402

View: 593

This book covers statistical inference based on the likelihood function. Discusses frequentist likelihood-based inference from a Fisherian viewpoint, Bayesian inference techniques including point and interval estimates, model choice and prediction and more.
Categories: Bayesian statistical decision theory

Introductory Statistical Inference with the Likelihood Function

Introductory Statistical Inference with the Likelihood Function

This book is about some of the basic principles of statistics that are necessary to understand and evaluate methods for analyzing complex data sets. The likelihood function is used for pure likelihood inference throughout the book.

Author: Charles A. Rohde

Publisher: Springer

ISBN: 9783319104614

Category: Medical

Page: 332

View: 483

This textbook covers the fundamentals of statistical inference and statistical theory including Bayesian and frequentist approaches and methodology possible without excessive emphasis on the underlying mathematics. This book is about some of the basic principles of statistics that are necessary to understand and evaluate methods for analyzing complex data sets. The likelihood function is used for pure likelihood inference throughout the book. There is also coverage of severity and finite population sampling. The material was developed from an introductory statistical theory course taught by the author at the Johns Hopkins University’s Department of Biostatistics. Students and instructors in public health programs will benefit from the likelihood modeling approach that is used throughout the text. This will also appeal to epidemiologists and psychometricians. After a brief introduction, there are chapters on estimation, hypothesis testing, and maximum likelihood modeling. The book concludes with sections on Bayesian computation and inference. An appendix contains unique coverage of the interpretation of probability, and coverage of probability and mathematical concepts.
Categories: Medical

Bayesian Statistics

Bayesian Statistics

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online.

Author: Source Wikipedia

Publisher: University-Press.org

ISBN: 1230616446

Category:

Page: 84

View: 837

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 83. Chapters: Bayesian probability, Prosecutor's fallacy, Likelihood function, Bayesian inference, Naive Bayes classifier, Bayesian network, Odds ratio, Variational Bayesian methods, Ensemble Kalman filter, Principle of maximum entropy, Bayesian spam filtering, Bayes estimator, Prior probability, Conjugate prior, Checking whether a coin is fair, Bayesian game, Imprecise probability, Data assimilation, Bayesian brain, Bayes factor, Graph cuts in computer vision, Jeffreys prior, Admissible decision rule, De Finetti's theorem, Bayesian inference in phylogeny, Maximum a posteriori estimation, Approximate Bayesian computation, Bayesian experimental design, Graphical model, Bayes linear statistics, Bayesian information criterion, Bayesian linear regression, Hierarchical Bayes model, Nested sampling algorithm, Evidence under Bayes theorem, Reference class problem, Recursive Bayesian estimation, Bayesian multivariate linear regression, Posterior probability, Credible interval, Extrapolation domain analysis, Hyperprior, Leonard Jimmie Savage, Deviance information criterion, AODE, Markov logic network, Bayesian search theory, Random naive Bayes, Bayesian average, A priori, Calibrated probability assessment, Hyperparameter, Gaussian process emulator, Marginal likelihood, GLUE, Aumann's agreement theorem, Precision, Base rate, Cromwell's rule, Speed prior, Bayesian econometrics, Expectation propagation, Strong prior, Sparse binary polynomial hashing, International Society for Bayesian Analysis.
Categories:

Bayesian Inference with Geodetic Applications

Bayesian Inference with Geodetic Applications

This introduction to Bayesian inference places special emphasis on applications.

Author: Karl-Rudolf Koch

Publisher: Springer

ISBN: 9783540466017

Category: Science

Page: 199

View: 945

This introduction to Bayesian inference places special emphasis on applications. All basic concepts are presented: Bayes' theorem, prior density functions, point estimation, confidence region, hypothesis testing and predictive analysis. In addition, Monte Carlo methods are discussed since the applications mostly rely on the numerical integration of the posterior distribution. Furthermore, Bayesian inference in the linear model, nonlinear model, mixed model and in the model with unknown variance and covariance components is considered. Solutions are supplied for the classification, for the posterior analysis based on distributions of robust maximum likelihood type estimates, and for the reconstruction of digital images.
Categories: Science

Comparative Statistical Inference

Comparative Statistical Inference

This book will be welcomed by both the student and practising statistician wishing to study at a fairly elementary level, the basic conceptual and interpretative distinctions between the different approaches, how they interrelate, what ...

Author: Vic Barnett

Publisher: John Wiley & Sons

ISBN: 9780470317792

Category: Mathematics

Page: 410

View: 423

This fully updated and revised third edition, presents a wide ranging, balanced account of the fundamental issues across the full spectrum of inference and decision-making. Much has happened in this field since the second edition was published: for example, Bayesian inferential procedures have not only gained acceptance but are often the preferred methodology. This book will be welcomed by both the student and practising statistician wishing to study at a fairly elementary level, the basic conceptual and interpretative distinctions between the different approaches, how they interrelate, what assumptions they are based on, and the practical implications of such distinctions. As in earlier editions, the material is set in a historical context to more powerfully illustrate the ideas and concepts. Includes fully updated and revised material from the successful second edition Recent changes in emphasis, principle and methodology are carefully explained and evaluated Discusses all recent major developments Particular attention is given to the nature and importance of basic concepts (probability, utility, likelihood etc) Includes extensive references and bibliography Written by a well-known and respected author, the essence of this successful book remains unchanged providing the reader with a thorough explanation of the many approaches to inference and decision making.
Categories: Mathematics

Bayesian Inference

Bayesian Inference

This new edition offers a comprehensive introduction to the analysis of data using Bayes rule.

Author: Hanns Ludwig Harney

Publisher: Springer

ISBN: 9783319416441

Category: Science

Page: 243

View: 698

This new edition offers a comprehensive introduction to the analysis of data using Bayes rule. It generalizes Gaussian error intervals to situations in which the data follow distributions other than Gaussian. This is particularly useful when the observed parameter is barely above the background or the histogram of multiparametric data contains many empty bins, so that the determination of the validity of a theory cannot be based on the chi-squared-criterion. In addition to the solutions of practical problems, this approach provides an epistemic insight: the logic of quantum mechanics is obtained as the logic of unbiased inference from counting data. New sections feature factorizing parameters, commuting parameters, observables in quantum mechanics, the art of fitting with coherent and with incoherent alternatives and fitting with multinomial distribution. Additional problems and examples help deepen the knowledge. Requiring no knowledge of quantum mechanics, the book is written on introductory level, with many examples and exercises, for advanced undergraduate and graduate students in the physical sciences, planning to, or working in, fields such as medical physics, nuclear physics, quantum mechanics, and chaos.
Categories: Science

Introduction to Bayesian Statistics

Introduction to Bayesian Statistics

"...this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels.

Author: William M. Bolstad

Publisher: John Wiley & Sons

ISBN: 9781118593226

Category: Mathematics

Page: 624

View: 122

"...this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels. It is a well-written book on elementary Bayesian inference, and the material is easily accessible. It is both concise and timely, and provides a good collection of overviews and reviews of important tools used in Bayesian statistical methods." There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. In this third Edition, four newly-added chapters address topics that reflect the rapid advances in the field of Bayesian statistics. The authors continue to provide a Bayesian treatment of introductory statistical topics, such as scientific data gathering, discrete random variables, robust Bayesian methods, and Bayesian approaches to inference for discrete random variables, binomial proportions, Poisson, and normal means, and simple linear regression. In addition, more advanced topics in the field are presented in four new chapters: Bayesian inference for a normal with unknown mean and variance; Bayesian inference for a Multivariate Normal mean vector; Bayesian inference for the Multiple Linear Regression Model; and Computational Bayesian Statistics including Markov Chain Monte Carlo. The inclusion of these topics will facilitate readers' ability to advance from a minimal understanding of Statistics to the ability to tackle topics in more applied, advanced level books. Minitab macros and R functions are available on the book's related website to assist with chapter exercises. Introduction to Bayesian Statistics, Third Edition also features: Topics including the Joint Likelihood function and inference using independent Jeffreys priors and join conjugate prior The cutting-edge topic of computational Bayesian Statistics in a new chapter, with a unique focus on Markov Chain Monte Carlo methods Exercises throughout the book that have been updated to reflect new applications and the latest software applications Detailed appendices that guide readers through the use of R and Minitab software for Bayesian analysis and Monte Carlo simulations, with all related macros available on the book's website Introduction to Bayesian Statistics, Third Edition is a textbook for upper-undergraduate or first-year graduate level courses on introductory statistics course with a Bayesian emphasis. It can also be used as a reference work for statisticians who require a working knowledge of Bayesian statistics.
Categories: Mathematics

A Student s Guide to Bayesian Statistics

A Student   s Guide to Bayesian Statistics

Supported by a wealth of learning features, exercises, and visual elements as well as online video tutorials and interactive simulations, this book is the first student-focused introduction to Bayesian statistics.

Author: Ben Lambert

Publisher: SAGE

ISBN: 9781526418265

Category: Reference

Page: 520

View: 947

Supported by a wealth of learning features, exercises, and visual elements as well as online video tutorials and interactive simulations, this book is the first student-focused introduction to Bayesian statistics. Without sacrificing technical integrity for the sake of simplicity, the author draws upon accessible, student-friendly language to provide approachable instruction perfectly aimed at statistics and Bayesian newcomers. Through a logical structure that introduces and builds upon key concepts in a gradual way and slowly acclimatizes students to using R and Stan software, the book covers: An introduction to probability and Bayesian inference Understanding Bayes' rule Nuts and bolts of Bayesian analytic methods Computational Bayes and real-world Bayesian analysis Regression analysis and hierarchical methods This unique guide will help students develop the statistical confidence and skills to put the Bayesian formula into practice, from the basic concepts of statistical inference to complex applications of analyses.
Categories: Reference

Practical Bayesian Inference

Practical Bayesian Inference

This book introduces the major concepts of probability and statistics, along with the necessary computational tools, for undergraduates and graduate students.

Author: Coryn A. L. Bailer-Jones

Publisher: Cambridge University Press

ISBN: 9781107192119

Category: Mathematics

Page: 295

View: 458

This book introduces the major concepts of probability and statistics, along with the necessary computational tools, for undergraduates and graduate students.
Categories: Mathematics

Topics on Methodological and Applied Statistical Inference

Topics on Methodological and Applied Statistical Inference

This book brings together selected peer-reviewed contributions from various research fields in statistics, and highlights the diverse approaches and analyses related to real-life phenomena.

Author: Tonio Di Battista

Publisher: Springer

ISBN: 9783319440934

Category: Mathematics

Page: 220

View: 154

This book brings together selected peer-reviewed contributions from various research fields in statistics, and highlights the diverse approaches and analyses related to real-life phenomena. Major topics covered in this volume include, but are not limited to, bayesian inference, likelihood approach, pseudo-likelihoods, regression, time series, and data analysis as well as applications in the life and social sciences. The software packages used in the papers are made available by the authors. This book is a result of the 47th Scientific Meeting of the Italian Statistical Society, held at the University of Cagliari, Italy, in 2014.
Categories: Mathematics

Likelihood Free Methods for Cognitive Science

Likelihood Free Methods for Cognitive Science

This book explains the foundation of approximate Bayesian computation (ABC), an approach to Bayesian inference that does not require the specification of a likelihood function.

Author: James J. Palestro

Publisher: Springer

ISBN: 9783319724256

Category: Psychology

Page: 129

View: 440

This book explains the foundation of approximate Bayesian computation (ABC), an approach to Bayesian inference that does not require the specification of a likelihood function. As a result, ABC can be used to estimate posterior distributions of parameters for simulation-based models. Simulation-based models are now very popular in cognitive science, as are Bayesian methods for performing parameter inference. As such, the recent developments of likelihood-free techniques are an important advancement for the field. Chapters discuss the philosophy of Bayesian inference as well as provide several algorithms for performing ABC. Chapters also apply some of the algorithms in a tutorial fashion, with one specific application to the Minerva 2 model. In addition, the book discusses several applications of ABC methodology to recent problems in cognitive science. Likelihood-Free Methods for Cognitive Science will be of interest to researchers and graduate students working in experimental, applied, and cognitive science.
Categories: Psychology

Towards Improved Model Evaluation

Towards Improved Model Evaluation

In the past decades, Bayesian methods have found widespread application and use in environmental systems modeling.

Author: Mojtaba Sadegh

Publisher:

ISBN: 1321564481

Category:

Page: 304

View: 235

In the past decades, Bayesian methods have found widespread application and use in environmental systems modeling. Bayes theorem states that the posterior probability of a hypothesis is proportional to the product of the prior probability of this hypothesis and the likelihood of the hypothesis given the new/incoming observations. In science and engineering, this hypothesis often constitutes some numerical simulation model, which summarizes using algebraic, empirical, and differential equations, state variables and fluxes, all our theoretical and/or practical knowledge of the system of interest, and unknown model parameters which are subject to inference using some data of the observed system response. The Bayesian approach is intimately related to the scientific method and uses an iterative cycle of hypothesis formulation (model), experimentation and data collection, and theory/hypothesis refinement to elucidate the rules that govern the natural world. Unfortunately, model refinement has proven to be very difficult in large part because of the poor diagnostic power of residual based likelihood functions. In this thesis, I present seven different chapters (publications) that introduce the theory, concepts and implementation of a diagnostic approach to model identification and evaluation. This approach, coined approximate Bayesian computation (ABC), relaxes the need for an explicit likelihood function in favor of one or more summary statistics, which when rooted in the relevant environmental theory have a much stronger and compelling diagnostic power than some average measure of the size of the error residuals. The proposed methodology is statistically coherent, and provides new insights into and guidance on model structural (epistemic) errors, model (hypothesis) refinement, system nonstationarity, and the information content of experimental data. What is more, the proposed diagnostic approach provides a much-needed statistical underpinning of the popular generalized likelihood uncertainty framework (GLUE) of Beven and co-workers, and presents a powerful alternative to subjective regularized inversion methods used in geophysical inversion. Finally, the DREAM(ABC) algorithm, developed to solve the diagnostic model evaluation problem, is orders of magnitude more efficient than commonly used ABC sampling methods, thereby permitting inference of parameter-rich system models.
Categories:

Introduction to Bayesian Statistics

Introduction to Bayesian Statistics

"...this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels.

Author: William M. Bolstad

Publisher: John Wiley & Sons

ISBN: 9781118091562

Category: Mathematics

Page: 624

View: 237

"...this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels. It is a well-written book on elementary Bayesian inference, and the material is easily accessible. It is both concise and timely, and provides a good collection of overviews and reviews of important tools used in Bayesian statistical methods." There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. In this third Edition, four newly-added chapters address topics that reflect the rapid advances in the field of Bayesian statistics. The authors continue to provide a Bayesian treatment of introductory statistical topics, such as scientific data gathering, discrete random variables, robust Bayesian methods, and Bayesian approaches to inference for discrete random variables, binomial proportions, Poisson, and normal means, and simple linear regression. In addition, more advanced topics in the field are presented in four new chapters: Bayesian inference for a normal with unknown mean and variance; Bayesian inference for a Multivariate Normal mean vector; Bayesian inference for the Multiple Linear Regression Model; and Computational Bayesian Statistics including Markov Chain Monte Carlo. The inclusion of these topics will facilitate readers' ability to advance from a minimal understanding of Statistics to the ability to tackle topics in more applied, advanced level books. Minitab macros and R functions are available on the book's related website to assist with chapter exercises. Introduction to Bayesian Statistics, Third Edition also features: Topics including the Joint Likelihood function and inference using independent Jeffreys priors and join conjugate prior The cutting-edge topic of computational Bayesian Statistics in a new chapter, with a unique focus on Markov Chain Monte Carlo methods Exercises throughout the book that have been updated to reflect new applications and the latest software applications Detailed appendices that guide readers through the use of R and Minitab software for Bayesian analysis and Monte Carlo simulations, with all related macros available on the book's website Introduction to Bayesian Statistics, Third Edition is a textbook for upper-undergraduate or first-year graduate level courses on introductory statistics course with a Bayesian emphasis. It can also be used as a reference work for statisticians who require a working knowledge of Bayesian statistics.
Categories: Mathematics

Essential Statistical Inference

Essential Statistical Inference

R code is woven throughout the text, and there are a large number of examples and problems. An important goal has been to make the topics accessible to a wide audience, with little overt reliance on measure theory.

Author: Dennis D. Boos

Publisher: Springer Science & Business Media

ISBN: 9781461448181

Category: Mathematics

Page: 568

View: 786

​This book is for students and researchers who have had a first year graduate level mathematical statistics course. It covers classical likelihood, Bayesian, and permutation inference; an introduction to basic asymptotic distribution theory; and modern topics like M-estimation, the jackknife, and the bootstrap. R code is woven throughout the text, and there are a large number of examples and problems. An important goal has been to make the topics accessible to a wide audience, with little overt reliance on measure theory. A typical semester course consists of Chapters 1-6 (likelihood-based estimation and testing, Bayesian inference, basic asymptotic results) plus selections from M-estimation and related testing and resampling methodology. Dennis Boos and Len Stefanski are professors in the Department of Statistics at North Carolina State. Their research has been eclectic, often with a robustness angle, although Stefanski is also known for research concentrated on measurement error, including a co-authored book on non-linear measurement error models. In recent years the authors have jointly worked on variable selection methods. ​
Categories: Mathematics

The Bayesian Inference Method and Its Application to Reliability Problems

The Bayesian Inference Method and Its Application to Reliability Problems

Statistical inference is the activity of characterizing the parameters of mathematical models by utilizing available sampling data.

Author: R. Lowell Smith

Publisher:

ISBN: OCLC:227553767

Category:

Page: 37

View: 452

Statistical inference is the activity of characterizing the parameters of mathematical models by utilizing available sampling data. This report discusses as a specific motivation the modeling of reliability problems and deals only with inference while avoiding the larger area of decision theory. The classical and Bayesian approaches to evaluating the parameter of the familiar exponential reliability model are compared. Classically, model parameters are unknown constants which can be estimated. From the Bayesian viewpoint model parameters are treated as distributed random variables. As is also ture of the classical maximum likelihood method, the determining or informational impact of the sampling data is represented completely by the likelihood function. Operationally, Bayesian inference involves applying Bayes theorem, a celecbrated consequence of conditional probability theory. The relevant probability background is developed and Bayes theorem derives. Bayesian inference has the very appealing capacity to incorporate previous information as well as current sampling inputs. Classical results are reproduced in the limiting forms of this involving noninformative prior distributions. Several application examples are discussed illustrating the use of both continuously and discretely distributed data and in one case emphasizing numerical methods.
Categories: