Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction to Bayesian Methods

1.1 Bayesian Methods: An Aerial Survey

1.2 Bayes’ Theorem

1.3 Bayes’ Theorem and the Focus Group

1.4 The Flavors of Probability

1.5 Summary

1.6 Notation Introduced in this Chapter

Chapter 2: A First Look at Bayesian Computation

2.1 Getting Started

2.2 Selecting the Likelihood Function

2.3 Selecting the Functional Form

2.4 Selecting the Prior

2.5 Finding the Normalizing Constant

2.6 Obtaining the Posterior

2.7 Communicating Findings

2.8 Predicting Future Outcomes

2.9 Summary

2.10 Exercises

2.11 Notation Introduced in this Chapter

Chapter 3: Computer-Assisted Bayesian Computation

3.1 Getting Started

3.2 Random Number Sequences

3.3 Monte Carlo Integration

3.4 Monte Carlo Simulation for Inference

3.5 The Conjugate Normal Model

3.6 In Practice: Inference for the Conjugate Normal Model

3.7 Count Data and the Conjugate Poisson Model

3.8 Summary

3.9 Exercises

3.10 Notation Introduced in this Chapter

3.11 Appendix—In Detail: Finding Posterior Distributions for the Normal Model

Chapter 4: Markov Chain Monte Carlo and Regression Models

4.1 Introduction to Markov Chain Monte Carlo

4.2 Fundamentals of MCMC

4.3 Gibbs Sampling

4.4 Gibbs Sampling and the Simple Linear Regression Model

4.5 In Practice: The Simple Linear Regression Model

4.6 The Metropolis Algorithm

4.7 Hastings’ Extension of the Metropolis Algorithm

4.8 Summary

4.9 Exercises

Chapter 5: Estimating Bayesian Models With WinBUGS

5.1 An Introduction to WinBUGS

5.2 In Practice: A First WinBUGS MODEL

5.3 In Practice: Models for the Mean in WinBUGS

5.4 Examining The Prior's Influence with Sensitivity Analysis

5.5 In Practice: Examining Proportions In WinBUGS

5.6 Analysis of Variance Models

5.7 Higher Order ANOVA Models

5.8 Regression and ANCOVA Models in WinBUGS

5.9 Summary

5.10 Chapter Appendix: Exporting WinBUGS MCMC Output TO R

5.11 Exercises

Chapter 6: Assessing Mcmc Performance in WinBUGS

6.1 Convergence Issues in MCMC Modeling

6.2 Output Diagnostics in WinBUGS

6.3 Reparameterizing to Improve Convergence

6.4 Number and Length of Chains

6.5 Metropolis–Hastings Acceptance Rates

6.6 Summary

6.7 Exercises

Chapter 7: Model Checking and Model Comparison

7.1 Graphical Model Checking

7.2 Predictive Densities and Checking Model Assumptions

7.3 Variable Selection Methods

7.5 Deviance Information Criterion

7.6 Summary

7.7 Exercises

Chapter 8: Hierarchical Models

8.1 Fundamentals of Hierarchical Models

8.2 The Random Coefficients Model

8.3 Hierarchical Models for Variance Terms

8.4 Functional Forms at Multiple Hierarchical Levels

8.5 In Detail: Modeling Covarying Hierarchical Terms

8.6 Summary

8.7 Exercises

8.8 Notation Introduced in this Chapter

Chapter 9: Generalized Linear Models

9.1 Fundamentals of Generalized Linear Models

9.2 Count Data Models: Poisson Regression

9.3 Models for Binary Data: Logistic Regression

9.4 The Probit Model

9.5 In Detail: Multinomial Logistic Regression for Categorical Outcomes

9.6 Hierarchical Models for Count Data

9.7 Hierarchical Models for Binary Data

9.8 Summary

9.9 Exercises

9.10 Notation Introduced in this Chapter

Chapter 10: Models For Difficult Data

10.1 Living with Outliers—Robust Regression Models

10.2 Handling Heteroscedasticity by Modeling Variance Parameters

10.3 Dealing with Missing Data

10.4 Types of Missing Data

10.5 Missing Covariate Data and Non-Normal Missing Data

10.6 Summary

10.7 Exercises

10.8 Notation Introduced in this Chapter

Chapter 11: Introduction To Latent Variable Models

11.1 Not Seen but Felt

11.2 Latent Variable Models for Binary Data

11.3 Structural Break Models

11.4 In Detail: The Ordinal Probit Model

11.5 Summary

11.6 Exercises

Appendix A: Common Statistical Distributions

References

Author Index

Subject Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Chapter 1: Introduction to Bayesian Methods

List of Illustrations

Figure 1.1

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

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Figure 4.9

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Figure 5.3

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Figure 5.6

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Figure 6.1

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Figure 6.3

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Figure 9.1

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Figure 9.3

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Figure 9.7

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Figure 10.1

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Figure 10.10

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 11.7

Figure 11.8

Figure 11.9

List of Tables

Table 1.1

Table 2.1

Table 3.1

Table 3.2

Table 4.1

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Table 5.7

Table 5.8

Table 5.9

Table 5.10

Table 5.11

Table 5.12

Table 5.13

Table 5.14

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Table 5.16

Table 5.17

Table 7.1

Table 7.2

Table 7.3

Table 7.4

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Table 7.6

Table 7.7

Table 8.1

Table 8.2

Table 8.3

Table 8.4

Table 8.5

Table 8.6

Table 8.7

Table 8.8

Table 9.1

Table 9.2

Table 9.3

Table 9.4

Table 9.5

Table 9.6

Table 9.7

Table 9.8

Table 9.9

Table 10.1

Table 10.2

Table 10.3

Table 10.4

Table 11.1

Table 11.2

Table A.1

Table A.2

Bayesian Methods for Management and Business

Pragmatic Solutions for Real Problems

Eugene D. Hahn

Department of Information and Decision Systems

Salisbury University

Salisbury, MD

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data:

Hahn, Eugene D.

Bayesian methods for management and business : pragmatic solutions for real problems / Eugene D. Hahn, Department of Information and Decision Systems, Salisbury University Salisbury, MD.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-63755-5 (hardback)

1. Management–Statistical methods. 2. Commercial statistics. 3. Bayesian statistical decision theory.

I. Title.

HD30.215.H34 2014

650.01′519542–dc23

2014011434

To Gene, Nora, David, Tara, and Nok.

Thank you for all that you do and have done. Also, thanks to Jeff Kottemann for

comments early in the book's development.

Preface

The use of Bayesian statistics has exploded over the past two decades. Researchers in many disciplines have used Bayesian statistics to reveal new insights and understand difficult problems. However, there are few discussions of Bayesian statistics that are focused on its practical applications to business and management. As a result, people with an orientation toward business and management problems were left to bridge the theory/application gap themselves with little guidance. In this book, we show how Bayesian statistics can help generate insights into business and management data. The book features a practical orientation with In Practice sections that go into detail the use of Bayesian approaches with actual business data.

On a personal note, I have spent countless hours to make this book, as much as possible, one that blends concepts and intuition along with mathematics. There is math in this book like in all statistically oriented books. However, it is my belief that an understanding of every formula is not required for getting something out of this book. As an analogy, you may have gone to a foreign country where you did not understand every word but were still able to get around. Here we will see that modern Bayesian methods use Monte Carlo Markov chain (MCMC) methods to perform much of the more difficult math involving calculus. Whenever you see the abbreviation MCMC, you can also think of it as the abbreviation for “Makes Calculus More Convenient” or “Makes Computation More Convenient”. MCMC makes these things more convenient because computer simulations handle the job for us. It is good to have a conceptual understanding of what an integration is for the few occasions that we discuss it in this book. But to summarize, an integral here is typically used like a weighted averaging technique to average out something that has variability. We are all very familiar with using summary numbers like average monthly sales to fill in for sales numbers that vary every month. When we integrate a metric over a probability distribution, we are doing something very similar. If you can remember this working definition, you will have a working understanding of the few formulas that mention calculus explicitly. The greatest amount of math appears in Chapters 3 and 4. Feel free to skip these entirely on a first reading and come back later at your leisure.

Instead of math, I think you should emphasize learning about modeling. A Bayesian model has three components. There is a likelihood, a functional form, and a set of priors. Being able to understand these three parts of a model and to choose them appropriately for your situation will be much more important in practice. For every model introduced, there is a detailed discussion in English about the considerations and choices that go into the model specification. Once we have gone through this discussion in English, we put the model in a compact notationthat summarizes the model. This summary is useful for describing your model to others as well as for writing programming code in the WinBUGS software. The WinBUGS software will handle estimation of your model using MCMC.

Like many business processes, Bayesian statistics is modular. Once you become familiar with the different kinds of models (modules), it is very easy to (re)assemble them into a structure that is applicable to your data. The customized model can then be estimated using MCMC. This characteristic of modern Bayesian methods has been quite important in its evolution to becoming an empirical method of choice in current statistical practice where customization to data characteristics or theoretical needs is important. After getting some experience, you will be in a position to come up with your own models that help you address your own business data.

Using the “In Practice” Sections

The In Practice sections provide worked examples using real data and WinBUGS code listings. The practically oriented reader can focus primarily on these sections to build up a library of Bayesian models for working with business data. Prior to each In Practice section, there is a discussion of conceptual material. However, different learning styles may benefit from different approaches. Therefore, feel free to consider reading and working with the In Practice material first if that better suits your learning style. Once you are comfortable with the In Practice material, you can then return to an examination of the more conceptual material.

Not all people are the same, so I have tried to make the book have something to offer to a variety of readers. Different pathways through the book can be used for people with different interests and different backgrounds. For MBA students with a solid grounding in the regression model and its prerequisites, the pathway might include Chapters 1, 2, 5–7, with other chapters being optional. For Masters of Science in Analytics students, the earlier parts of Chapters 8 and 9 could be added. Readers who would prefer a less technical, applications-oriented approach could cover Chapters 1, 2, 5 and 6, then focus on In Practice sections in Chapters 7 through 11. A solid grounding in the foundations of Bayesian inference (Chapters 3 and 4) will be helpful for those who want to make the most of Bayesian inference. A traditional approach would place foundational concepts early in the process and the book follows this approach. However, these concepts can be postponed for readers who are more interested in immediate solutions.

This book makes use of the freely available software WinBUGS and R. Code listings appear throughout the book. The code listings are available at the author's Web site at http://faculty.salisbury.edu/∼edhahn.

Using the “In Detail” Sections

The book also features In Detail sections where somewhat more advanced material is presented. These sections are for readers who would like to drill down more into a topic or who are looking for additional skill-building. Again, you (the reader) get to choose what you would like to learn about, and at what time. The In Detail sections can be read in their original sequence, or you can return to them as the need/interest arises.

Chapter 1
Introduction to Bayesian Methods

1.1 Bayesian Methods: An Aerial Survey

The modern business environment is awash in data. As a result, managers seek ways to summarize and simplify data to emphasize a select number of key aspects. They may also wish to examine whether certain kinds of structure and patterns are present in the data. Or, they may wish to use data to draw conclusions about other kinds of unobserved or latent phenomena they believe exist with respect to their businesses, customers, materials, and so on. These kinds of activities managers undertake are not mutually exclusive, but rather emphasize different aspects of the data discovery process.

Statistical methods are some of the most widely-used methods for data discovery. Many managers who have gone through an undergraduate or graduate business education will have encountered some of these methods. The methods that are typically taught to managers are called classical statistical methods. Classical methods are also known as frequentist methods because they derive from a frequency-based view of probability. Classical methods can be summarized as statistical methods that can be arrived at based on consideration of the likelihood function alone (Fisher, 1922). These include the familiar c01-math-0001-test, simple linear regression, and logit analysis by maximum likelihood. The likelihood function can be thought of as the “data function” since it quantifies the relative likelihood of parameter values in the data we have observed. The likelihood function allows the manager to summarize or estimate unknown parameters such as the lifetime value of a customer or the average failure rate of an important component.

In addition to the likelihood function, Bayesian methods also incorporate a prior distribution for parameters. If the manager has preexisting beliefs about parameters, such as an intuition about the average failure rate of a component, he/she can often express this in terms of a probability distribution. Having done so, Bayesian methods can proceed and the final results (summarized by the posterior distribution) will contain a blend of the information arising from the prior (i.e., from personal beliefs) and from the likelihood function (i.e., from the data). This blend will reflect the relative weight of information arising from the two sources. For example, if the prior is extremely concentrated, very large quantities of discrepant data will be required in order to produce a substantive change in the parameter estimate. No doubt you have met someone at some point in your life whose prior beliefs about something were rather difficult to change even with substantial evidence, and this is conceivably possible with the Bayesian approach as well. More commonly, however, a manager or researcher willadopt what is called a . Such priors are designed to have little effect on the conclusions that would be drawn from the data, and hence reflect an “open-mindedness” about the data in the sense that a broad range of values would be considered reasonably possible. Given the practical emphasis of business and management, these diffuse priors are almost always employed when one is interested in understanding the data because there is little point in analyzing data if we already wish to retain our preconceived notions. However, there are situations where one needs to be more careful about the impact of the prior, and the most common of these is the situation where the sample size is small. Since the posterior distribution reflects the relative weight of the data and the prior, we must exercise care when the data influence is light due to its scarcity (of course, we would also want to draw conclusions cautiously if classical methods were used with small samples).

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