Hierarchical structures and posterior probabilities

Models for Ecological Data: An Introduction. Author James S. Clark. Edition First. Publisher Princeton University Press. Pages 632. Price £38.95/£11.95. ISBN 9780691121789/2625.

May 22, 2008

The year 1761 saw the marriage (and subsequent coronation) of King George III and Duchess Sophia Charlotte of Mecklenburg-Strelitz. It also saw the death of Thomas Bayes.

Bayes was a Presbyterian minister in Tunbridge Wells in the 18th century during its height as a centre for fashion, elegance and exuberance. Like many clergy (such as Gregor Mendel and Gilbert White) he engaged in secular activities and his predilection was mathematics. He offered (although not published in his lifetime) a fantastic solution to problems associated with probabilities. Bayes conjectured a method for determining the probability of an event occurring depending on whether another event had occurred. This approach ranks as one of the most famous in mathematics and still remains controversial in modern-day statistical thinking.

Bayesian statistical analyses have grown in application and diversity since the mid-part of the last century, particularly with the advent of modern computer technologies. While not exclusively about Bayes and the Bayseian approach, Models for Ecological Data: An Introduction illustrates how we can adopt Bayes's approach for data analysis and statistical inference for ecological and environmental problems. Often the adoption of a Bayesian or classic (frequentist) approach to statistical analysis is plagued with prejudice and philosophical wranglings over the benefits (or otherwise) of one approach over another. Obviously as good scientists we should arm ourselves with the diverse array of approaches, selecting those that are most appropriate for the problem at hand. This text, aimed at upper undergraduates, postgraduates and researchers in the environmental sciences, emphasises how we might do this and shift between approaches as the nature of our environmental problems increase in complexity.

This book aims to introduce to a non-statistical audience approaches of statistical model development, inference and prediction. Since the revolutionary work by R. A. Fisher, experimental design and the comparison of models to data has been the mainstay of the methods of inference in the environmental sciences. This has two aims: to understand and to predict. As such, Models for Ecological Data: An Introduction is divided into four main sections: an introduction, elements of inference, larger models and more advanced methods.

In the section "Under elements of inference", Clark shows us lucidly how to develop statistical approaches for point estimates for simple population models, the elements of the Bayesian approach, appropriate confidence envelopes and prediction intervals and how to assess the goodness of fit of our models. This sets out the ground work for the development of further topics (hierarchical models, time-series models, analysis of spatial patterns) presented in subsequent chapters.

One of the central themes introduced into the book (under "Larger models") is the use of hierarchical models and structures. The Bayesian approach follows from the premise of a conditional probability. It depends on beliefs about unknown quantities in data (prior probabilities), modifying them in light of data (the likelihood function) to arrive at posterior probabilities. Post is prior times likelihood. With hierarchical models we have more structure in our prior beliefs.

Clark introduces these hierarchical structures carefully and thoughtfully through examples associated with differential mortality rates in subdivided populations. The chapter on "Hierarchical structures" extends this theme of hierarchical models deep into problems where a Bayesian approach is entirely appropriate.

The final chapters of the book focus on the analysis of data collected through time, across space or both. Again, these chapters begin with carefully presented problems and extend the analysis to more detailed ecological questions where Bayesian approaches can be appropriately implemented.

This is a wonderful book in which Clark illustrates the range of statistical tools available for data analysis and model comparisons to data. The fact that we are, arguably, all Bayesians in our probabilistic perception of the world emphasises the need that a book like this should find its way into the hands of all those who seek a rigorous understanding of the natural world.

Who is it for? The book is aimed at upper undergraduates (maths and biology), postgraduates and researchers in the environmental sciences. But it should be available to all those seeking solutions to environmental problems.

Presentation - Laid out in four sections (introduction, elements of inference, larger models, more advanced methods), each has both theoretical background and practical implementation.

Would you recommend it? Definitely - it is a must-have for those interested in understanding and developing the analysis of their environmental data.

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