Bayesian modeling has become an indispensable tool for ecological research because it is uniquely suited to deal with complexity in a statistically coherent way. Bayesian Models provides a comprehensive and accessible introduction to the latest Bayesian methods – in language ecologists can understand. Unlike other books on the subject, this one emphasizes the principles behind the computations, giving ecologists a big-picture understanding of how to implement this powerful statistical approach.
Bayesian Models is an essential primer for non-statisticians. It begins with a definition of probability and develops a step-by-step sequence of connected ideas, including basic distribution theory, network diagrams, hierarchical models, Markov chain Monte Carlo, and inference from single and multiple models. This unique book places less emphasis on computer coding, favoring instead a concise presentation of the mathematical statistics needed to understand how and why Bayesian analysis works. It also explains how to write out properly formulated hierarchical Bayesian models and use them in computing, research papers, and proposals. This primer enables ecologists to understand the statistical principles behind Bayesian modeling and apply them to research, teaching, policy, and management.
I Fundamentals 1
1 PREVIEW 3
1.1 A Line of Inference for Ecology 4
1.2 An Example Hierarchical Model 11
1.3 What Lies Ahead? 15
2 DETERMINISTIC MODELS 17
2.1 Modeling Styles in Ecology 17
2.2 A Few Good Functions 21
3 PRINCIPLES OF PROBABILITY 29
3.1 Why Bother with First Principles? 29
3.2 Rules of Probability 31
3.3 Factoring Joint Probabilities 36
3.4 Probability Distributions 39
4 LIKELIHOOD 71
4.1 Likelihood Functions 71
4.2 Likelihood Profiles 74
4.3 Maximum Likelihood 76
4.4 The Use of Prior Information in Maximum Likelihood 77
5 SIMPLE BAYESIAN MODELS 79
5.1 Bayes' Theorem 81
5.2 The Relationship between Likelihood and Bayes' 85
5.3 Finding the Posterior Distribution in Closed Form 86
5.4 More about Prior Distributions 90
6 HIERARCHICAL BAYESIAN MODELS 107
6.1 What Is a Hierarchical Model? 108
6.2 Example Hierarchical Models 109
6.3 When Are Observation and Process Variance Identifiable? 141
II Implementation 143
7 MARKOV CHAIN MONTE CARLO 145
7.1 Overview 145
7.2 How Does MCMC Work? 146
7.3 Specifics of the MCMC Algorithm 150
7.4 MCMC in Practice 177
8 INFERENCE FROM A SINGLE MODEL 181
8.1 Model Checking 181
8.2 Marginal Posterior Distributions 190
8.3 Derived Quantities 194
8.4 Predictions of Unobserved Quantities 196
8.5 Return to the Wildebeest 201
9 INFERENCE FROM MULTIPLE MODELS 209
9.1 Model Selection 210
9.2 Model Probabilities and Model Averaging 222
9.3 Which Method to Use? 227
III Practice in Model Building 231
10 WRITING BAYESIAN MODELS 233
10.1 A General Approach 233
10.2 An Example of Model Building: Aboveground Net Primary Production in Sagebrush Steppe 237
11 PROBLEMS 243
11.1 Fisher's Ticks 244
11.2 Light Limitation of Trees 245
11.3 Landscape Occupancy of Swiss Breeding Birds 246
11.4 Allometry of Savanna Trees 247
11.5 Movement of Seals in the North Atlantic 248
12 SOLUTIONS 251
12.1 Fisher's Ticks 251
12.2 Light Limitation of Trees 256
12.3 Landscape Occupancy of Swiss Breeding Birds 259
12.4 Allometry of Savanna Trees 264
12.5 Movement of Seals in the North Atlantic 268
A Probability Distributions and Conjugate Priors 279
N. Thompson Hobbs is senior research scientist at the Natural Resource Ecology Laboratory and professor in the Department of Ecosystem Science and Sustainability at Colorado State University.
Mevin B. Hooten is associate professor in the Department of Fish, Wildlife, and Conservation Biology and the Department of Statistics at Colorado State University, and assistant unit leader in the US Geological Survey's Colorado Cooperative Fish and Wildlife Research Unit.
"This pitch-perfect exposition shows how Bayesian modeling can be used to quantify our uncertain world. Ecologists – and for that matter, scientists everywhere – are aware of these uncertainties, and this book gives them the understanding to do something about it. Hobbs and Hooten take us on a signposted journey through the culture, construction, and consequences of conditional-probability modeling, readying us to take our own scientific journeys through uncertain landscapes."
– Noel Cressie, University of Wollongong, Australia
"Hobbs and Hooten provide a complete guide to Bayesian thinking and statistics. This is a book by ecologists for ecologists. One of the powers of Bayesian thinking is how it enables you to evaluate knowledge accumulated through multiple experiments and publications, and this excellent primer provides a firm grounding in the hierarchical models that are now the standard approach to evaluating disparate data sets."
– Ray Hilborn, University of Washington
"In this uniquely well-written and accessible text, Hobbs and Hooten show how to think clearly in a Bayesian framework about data, models, and linking data with models. They provide the necessary tools to develop, implement, and analyze a wide range of ecologically interesting models. There's something new and exciting in this book for every practicing ecologist."
– Aaron M. Ellison, Harvard University
"Hobbs and Hooten provide an important bridge between standard statistical texts and more advanced Bayesian books, even those aimed at ecologists. Ecological models are complex. Building from likelihood to simple and hierarchical Bayesian models, the authors do a superb job of focusing on concepts, from philosophy to the necessary mathematical and statistical tools. This practical and understandable book belongs on the shelves of all scientists and statisticians interested in ecology."
– Jay M. Ver Hoef, Statistician, NOAA-NMFS Alaska Fisheries Science Center
"Tackling an important and challenging topic, Hobbs and Hooten provide non-statistically-trained ecologists with the skills they need to use hierarchical Bayesian models accurately and comfortably. The combination of technical explanations and practical examples is great. This book is a valuable contribution that will be widely used."
– Benjamin Bolker, McMaster University
"This excellent book is one of the best-written and most complete primers on Bayesian hierarchical modeling I have seen. Hobbs and Hooten anticipate many of the common pitfalls and concerns that arise when non-statisticians are introduced to this material. Researchers across a wide range of disciplines will find this book valuable."
– Christopher Wikle, University of Missouri