Bayesian Analysis of Stochastic Process Models - David Insua - ebook

Bayesian Analysis of Stochastic Process Models ebook

David Insua

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Opis

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.

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Liczba stron: 458




Contents

Cover

Series

Title Page

Copyright

Preface

Part One: Basic Concepts and Tools

1: Stochastic processes

1.1 Introduction

1.2 Key concepts in stochastic processes

1.3 Main classes of stochastic processes

1.4 Inference, prediction, and decision-making

1.5 Discussion

References

2: Bayesian analysis

2.1 Introduction

2.2 Bayesian statistics

2.3 Bayesian decision analysis

2.4 Bayesian computation

2.5 Discussion

References

Part Two: Models

3: Discrete time Markov chains and extensions

3.1 Introduction

3.2 Important Markov chain models

3.3 Inference for first-order, time homogeneous, Markov chains

3.4 Special topics

3.5 Case study: Wind directions at Gijón

3.6 Markov decision processes

3.7 Discussion

References

4: Continuous time Markov chains and extensions

4.1 Introduction

4.2 Basic setup and results

4.3 Inference and prediction for CTMCs

4.4 Case study: Hardware availability through CTMCs

4.5 Semi-Markovian processes

4.6 Decision-making with semi-Markovian decision processes

4.7 Discussion

References

5: Poisson processes and extensions

5.1 Introduction

5.2 Basics on Poisson processes

5.3 Homogeneous Poisson processes

5.4 Nonhomogeneous Poisson processes

5.5 Compound Poisson processes

5.6 Further extensions of Poisson processes

5.7 Case study: Earthquake occurrences

5.8 Discussion

References

6: Continuous time continuous space processes

6.1 Introduction

6.2 Gaussian processes

6.3 Brownian motion and FBM

6.4 Diffusions

6.5 Case study: Predator–prey systems

6.6 Discussion

References

Part Three: Applications

7: Queueing analysis

7.1 Introduction

7.2 Basic queueing concepts

7.3 The main queueing models

7.4 Bayesian inference for queueing systems

7.5 Bayesian inference for the system

7.6 Inference for non-Markovian systems

7.7 Decision problems in queueing systems

7.8 Case study: Optimal number of beds in a hospital

7.9 Discussion

References

8: Reliability

8.1 Introduction

8.2 Basic reliability concepts

8.3 Renewal processes

8.4 Poisson processes

8.5 Other processes

8.6 Maintenance

8.7 Case study: Gas escapes

8.8 Discussion

References

9: Discrete event simulation

9.1 Introduction

9.2 Discrete event simulation methods

9.3 A Bayesian view of DES

9.4 Case study: A queueing system

9.5 Bayesian output analysis

9.6 Simulation and optimization

9.7 Discussion

References

10: Risk analysis

10.1 Introduction

10.2 Risk measures

10.3 Ruin problems

10.4 Case study: Estimation of finite-time ruin probabilities in the Sparre Andersen model

10.5 Discussion

References

Appendix A: Main distributions

Discrete distributions

Continuous distributions

Multivariate distributions

References

Appendix B: Generating functions and the Laplace–Stieltjes transform

Probability generating function

Moment generating function

Laplace–Stieltjes transform

References

Index

Series List

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: David J. Balding, Noel A.C. Cressie, Garrett M. Fitzmaurice, Harvey Goldstein, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F.M. Smith, Ruey S. Tsay, Sanford Weisberg

Editors Emeriti: Vic Barnett, Ralph A. Bradley, J. Stuart Hunter, J.B. Kadane, David G. Kendall, Jozef L. Teugels

A complete list of the titles in this series appears at the end of this volume.

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Library of Congress Cataloging-in-Publication Data

Ruggeri, Fabrizio. Bayesian analysis of stochastic process models / Fabrizio Ruggeri, Michael P. Wiper, David Rios Insua. pages cm Includes bibliographical references and index. ISBN 978-0-470-74453-6 (hardback) 1. Bayesian statistical decision theory. 2. Stochastic processes. I. Wiper, Michael P. II. Rios Insua, David, 1964– III. Title. QA279.5.R84 2012 519.5′42–dc23 2012000092

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-74453-6

Preface

To the best of our knowledge, this is the first book focusing on Bayesian analysis of stochastic process models at large. We believe that recent developments in the field and the growing interest in this topic deserve a book-length treatment.

The advent of cheap computing power and the developments in Markov chain Monte Carlo simulation produced a revolution within the field of Bayesian statistics around the beginning of the 1990s, allowing a true ‘model liberation’ that permitted treating models that previously we could only dream of dealing with. This has challenged analysts in trying to deal with more complex problems. Given this great advance in computing power, it is no surprise that several researchers have attempted to deal with stochastic processes in a Bayesian fashion, moving away from the usual assumptions of independent and identically distributed (IID) data. In 1998, this led us to organize the first Workshop on Bayesian Analysis of Stochastic Processes in Madrid. The seventh edition of this conference was held in 2011, which is an illustration of the great current interest in this subject area. Given the numerous papers written, we felt, therefore, that the time was right to provide a systematic account of developments in Bayesian analysis of stochastic processes. In doing this, it is interesting to note that most books in stochastic processes have referred mainly to probabilistic aspects and there are many fewer texts that treat them from a (classical) statistical perspective.

In this monograph, we have emphasized five salient aspects:

1. The use of Bayesian methods as a unifying scientific paradigm.
2. Forecasting and decision-making problems, going beyond the usual focus on inference.
3. Computational tools that facilitate dealing with complex stochastic process based models.
4. The applicability of results. We include at least one real case study per chapter. These examples come from engineering, business, geosciences and biology contexts, showing the broad spectrum of possible applications.
5. Ample references and bibliographic discussions are provided at the end of each chapter, so that readers may pursue a more in-depth study of the corresponding topics and identify challenging areas for new research.

Our monograph is structured in three parts:

1. Part One refers to basic concepts and tools both in stochastic processes and Bayesian analysis. We review key probabilistic results and tools to deal with stochastic processes, the definitions of the key processes that we shall face and we set up the basic inference, prediction, and decision-making problems in relation with stochastic processes. We then review key results in Bayesian analysis that we shall use later on, with emphasis on computations and decision-analytic issues.
2. Part Two illustrates Bayesian analysis of some of the key stochastic process models. This section consists of four chapters. The first two are devoted to Markov chains and extensions in discrete time and continuous time, respectively. The third chapter contains a detailed analysis of Poisson processes, with particular emphasis on nonhomogeneous ones. Finally, the fourth chapter deals with continuous time/continuous space processes, in particular Gaussian processes and diffusions.
3. Part Three also contains four chapters. These refer to application areas in which there are several interrelated processes that make the situations analyzed more complex. The first chapter refers to queueing models that include arrival and service processes. The second chapter refers to reliability problems that include failure and repair processes. The third provides a framework for Bayesian analysis of extremely complex models that can only be described through discrete event simulation models. Finally, we provide an approach to some problems in Bayesian risk analysis.

We are grateful to the many institutions that have supported at various points our research in this field. In particular, D.R.I. wants to acknowledge the Spanish Ministry of Science and Education (eColabora and Riesgos), the Spanish Ministry of Industry, the Government of Madrid through the Riesgos-CM program, the European Science Foundation through the ALGODEC program, the SECONOMICS project, the Statistical and Applied Mathematical Sciences Institute, Apara Software and MTP. F.R. wants to acknowledge the Statistical and Applied Mathematical Sciences Institute. M.P.W. wishes to acknowledge support from projects of the Spanish Ministry of Science and Education and the Government of Madrid.

We have also benefited of our collaboration in these areas over various years with many colleagues and former students. Specifically, D.R.I. would like to thank Javier Cano, Jesus Ríos, Miguel Herrero, Javier Girón, Concha Bielza, Peter Müller, Javier Moguerza, Dipak Dey, Mircea Grigoriu, Jim Berger, Armi Moreno, Simon French, Jacinto Martin, David Banks, Raquel Montes, and Miguel Virto. He specially misses many hours of discussion and collaboration with Sixto Ríos, Sixto Ríos Insua, and Jorge Muruzábal. F.R. would like to thank Sara Pasquali, Antonio Pievatolo, Renata Rotondi, Bruno Betrò, Refik Soyer, Siva Sivaganesan, Gianni Gilioli, Fernanda D’Ippoliti, Cristina Mazzali, Loretta Masini, Emanuela Saccuman, Davide Cavallo, Franco Caron, Enrico Cagno, and Mauro Mancini. Finally, M.P.W. has been much helped by Andrés Alonso, Conchi Ausín, Carmen Broto, José Antonio Carnicero, Pedro Galeano, Cristina García, Ana Paula Palacios, Pepa Rodríguez-Cobo, and Nuria Torrado.

The patience and competence of the personnel at Wiley, and in particular of Richard Davies and Heather Kay, is heartily appreciated, as well as the support from Kathryn Sharples and Ilaria Meliconi who played a fundamental role in the start of this project.

Last, but not least, our families (Susana, Isa, and Ota; Anna, Giacomo, and Lorenzo; Imogen, Pike†, and Bo) have provided us with immense support and the required warmth to complete this long-lasting project.

Valdoviño, Milano, and Getafe November 2011

Part One

BASIC CONCEPTS AND TOOLS

2

Bayesian analysis

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