Discretization And Mcmc Convergence Assessment
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Discretization And Mcmc Convergence Assessment
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Author : Christian P. Robert
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-08-13
Discretization And Mcmc Convergence Assessment written by Christian P. Robert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-08-13 with Mathematics categories.
This monograph proposes several approaches to convergence monitoring for MCMC algorithms which are centered on the theme of discrete Markov chains. After a short introduction to MCMC methods, including recent developments like perfect simulation and Langevin Metropolis-Hastings algorithms, and to the current convergence diagnostics, the contributors present the theoretical basis for a study of MCMC convergence using discrete Markov chains and their specificities. The contributors stress in particular that this study applies in a wide generality, starting with latent variable models like mixtures, then extending the scope to chains with renewal properties, and concluding with a general Markov chain. They then relate the different connections with discrete or finite Markov chains with practical convergence diagnostics which are either graphical plots (allocation map, divergence graph, variance stabilizing, normality plot), stopping rules (normality, stationarity, stability tests), or confidence bounds (divergence, asymptotic variance, normality). Most of the quantitative tools take advantage of manageable versions of the CLT. The different methods proposed here are first evaluated on a set of benchmark examples and then studied on three full scale realistic applications, along with the standard convergence diagnostics: A hidden Markov modelling of DNA sequences, including a perfect simulation implementation, a latent stage modelling of the dynamics of HIV infection, and a modelling of hospitalization duration by exponential mixtures. The monograph is the outcome of a monthly research seminar held at CREST, Paris, since 1995. The seminar involved the contributors to this monograph and was led by Christian P. Robert, Head of the Satistics Laboratory at CREST and Professor of Statistics at the University of Rouen since 1992.
Discretization And Mcmc Convergence Assessment
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Author : Christian P. Robert
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Discretization And Mcmc Convergence Assessment written by Christian P. Robert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The exponential increase in the use of MCMC methods and the corre sponding applications in domains of even higher complexity have caused a growing concern about the available convergence assessment methods and the realization that some of these methods were not reliable enough for all-purpose analyses. Some researchers have mainly focussed on the con vergence to stationarity and the estimation of rates of convergence, in rela tion with the eigenvalues of the transition kernel. This monograph adopts a different perspective by developing (supposedly) practical devices to assess the mixing behaviour of the chain under study and, more particularly, it proposes methods based on finite (state space) Markov chains which are obtained either through a discretization of the original Markov chain or through a duality principle relating a continuous state space Markov chain to another finite Markov chain, as in missing data or latent variable models. The motivation for the choice of finite state spaces is that, although the resulting control is cruder, in the sense that it can often monitor con vergence for the discretized version alone, it is also much stricter than alternative methods, since the tools available for finite Markov chains are universal and the resulting transition matrix can be estimated more accu rately. Moreover, while some setups impose a fixed finite state space, other allow for possible refinements in the discretization level and for consecutive improvements in the convergence monitoring.
Extreme Value Modeling And Risk Analysis
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Author : Dipak K. Dey
language : en
Publisher: CRC Press
Release Date : 2016-01-06
Extreme Value Modeling And Risk Analysis written by Dipak K. Dey and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-01-06 with Mathematics categories.
Extreme Value Modeling and Risk Analysis: Methods and Applications presents a broad overview of statistical modeling of extreme events along with the most recent methodologies and various applications. The book brings together background material and advanced topics, eliminating the need to sort through the massive amount of literature on the subje
Nonlinear Dynamics And Statistics
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Author : Alistair I. Mees
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Nonlinear Dynamics And Statistics written by Alistair I. Mees and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Business & Economics categories.
All models are lies. "The Earth orbits the sun in an ellipse with the sun at one focus" is false, but accurate enough for almost all purposes. This book describes the current state of the art of telling useful lies about time-varying systems in the real world. Specifically, it is about trying to "understand" (that is, tell useful lies about) dynamical systems directly from observa tions, either because they are too complex to model in the conventional way or because they are simply ill-understood. B(:cause it overlaps with conventional time-series analysis, building mod els of nonlinear dynamical systems directly from data has been seen by some observers as a somewhat ill-informed attempt to reinvent time-series analysis. The truth is distinctly less trivial. It is surely impossible, except in a few special cases, to re-create Newton's astonishing feat of writing a short equation that is an excellent description of real-world phenomena. Real systems are connected to the rest of the world; they are noisy, non stationary, and have high-dimensional dynamics; even when the dynamics contains lower-dimensional attractors there is almost never a coordinate system available in which these at tractors have a conventionally simple description.
The Bayesian Choice
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Author : Christian Robert
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-08-27
The Bayesian Choice written by Christian Robert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-08-27 with Mathematics categories.
This is an introduction to Bayesian statistics and decision theory, including advanced topics such as Monte Carlo methods. This new edition contains several revised chapters and a new chapter on model choice.
Nonparametric Monte Carlo Tests And Their Applications
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Author : Li-Xing Zhu
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-08-09
Nonparametric Monte Carlo Tests And Their Applications written by Li-Xing Zhu and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-09 with Mathematics categories.
Monte Carlo approximation to the null distribution of the test provides a convenient means of testing model fit. This book proposes a Monte Carlo-based methodology to construct this type of approximation when the model is semistructured. It addresses both applied and theoretical aspects of nonparametric Monte Carlo tests.
Robust Bayesian Analysis
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Author : David Rios Insua
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Robust Bayesian Analysis written by David Rios Insua and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Robust Bayesian analysis aims at overcoming the traditional objection to Bayesian analysis of its dependence on subjective inputs, mainly the prior and the loss. Its purpose is the determination of the impact of the inputs to a Bayesian analysis (the prior, the loss and the model) on its output when the inputs range in certain classes. If the impact is considerable, there is sensitivity and we should attempt to further refine the information the incumbent classes available, perhaps through additional constraints on and/ or obtaining additional data; if the impact is not important, robustness holds and no further analysis and refinement would be required. Robust Bayesian analysis has been widely accepted by Bayesian statisticians; for a while it was even a main research topic in the field. However, to a great extent, their impact is yet to be seen in applied settings. This volume, therefore, presents an overview of the current state of robust Bayesian methods and their applications and identifies topics of further in terest in the area. The papers in the volume are divided into nine parts covering the main aspects of the field. The first one provides an overview of Bayesian robustness at a non-technical level. The paper in Part II con cerns foundational aspects and describes decision-theoretical axiomatisa tions leading to the robust Bayesian paradigm, motivating reasons for which robust analysis is practically unavoidable within Bayesian analysis.
Nonparametric Goodness Of Fit Testing Under Gaussian Models
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Author : Yuri Ingster
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-11-12
Nonparametric Goodness Of Fit Testing Under Gaussian Models written by Yuri Ingster and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-11-12 with Mathematics categories.
This book presents the modern theory of nonparametric goodness-of-fit testing. It fills the gap in modern nonparametric statistical theory by discussing hypothesis testing and addresses mathematical statisticians who are interesting in the theory of non-parametric statistical inference. It will be of interest to specialists who are dealing with applied non-parametric statistical problems relevant in signal detection and transmission and in technical and medical diagnostics among others.
Bayesian Missing Data Problems
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Author : Ming T. Tan
language : en
Publisher: CRC Press
Release Date : 2009-08-26
Bayesian Missing Data Problems written by Ming T. Tan and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-08-26 with Mathematics categories.
Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation presents solutions to missing data problems through explicit or noniterative sampling calculation of Bayesian posteriors. The methods are based on the inverse Bayes formulae discovered by one of the author in 1995. Applying the Bayesian approach to important real-wor
Lectures On Monte Carlo Methods
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Author : Neal Noah Madras
language : en
Publisher: American Mathematical Soc.
Release Date : 2002
Lectures On Monte Carlo Methods written by Neal Noah Madras and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
Monte Carlo methods form an experimental branch of mathematics that employs simulations driven by random number generators. These methods are often used when others fail, since they are much less sensitive to the ``curse of dimensionality'', which plagues deterministic methods in problems with a large number of variables. Monte Carlo methods are used in many fields: mathematics, statistics, physics, chemistry, finance, computer science, and biology, for instance. This book is an introduction to Monte Carlo methods for anyone who would like to use these methods to study various kinds of mathematical models that arise in diverse areas of application. The book is based on lectures in a graduate course given by the author. It examines theoretical properties of Monte Carlo methods as well as practical issues concerning their computer implementation and statistical analysis. The only formal prerequisite is an undergraduate course in probability. The book is intended to be accessible to students from a wide range of scientific backgrounds. Rather than being a detailed treatise, it covers the key topics of Monte Carlo methods to the depth necessary for a researcher to design, implement, and analyze a full Monte Carlo study of a mathematical or scientific problem. The ideas are illustrated with diverse running examples. There are exercises sprinkled throughout the text. The topics covered include computer generation of random variables, techniques and examples for variance reduction of Monte Carlo estimates, Markov chain Monte Carlo, and statistical analysis of Monte Carlo output.