



Review by Eddie Shoesmith, University of Buckingham, in The Statistician, No. 47, Part 4, 1998. 'In this book, I present a new mathematical and philosophical foundation for probability and show how this foundation can help us to understand causality.' This first sentence from the preface is an indication of how ambitious Shafer's book is. The basis for the new foundation is the (hitherto humble) probability tree. His aim is to demonstrate that this can be used, in place of sample or outcome space, as an alternative startingpoint for the development of insightful and applicable probabilistic ideas. In particular, he sets out to show that, when used in conjunction with a number of associated new concepts, the probability tree approach advances discussion of 'causality' in circumstances where the unfolding of events in time is contingent in nature. A later sentence from the preface gives a pointer to one of the stimuli for Shafer's work, and to the area where his new foundations are likely to find most immediate development. 'This book has been inspired by the debate about causal reasoning in artificial intelligence, and it provides a foundation that will facilitate the further development of probabilistic expert systems.' In Shafer's new approach, probability trees represent 'nature's' (an omnipotent observer's) partial knowledge. Taking the probability tree as the foundation for developing probabilistic ideas, he argues, alleviates the philosophical and interpretational difficulties that frequentists and Bayesians face. He contends that the new foundation integrates 'frequency and belief into a single story, in which an observer's knowledge develops step by step'. The author nominates Chapters 14 and 15, which occupy about 60 pages towards the end of the main text, as the 'heart' of the book. They deal respectively with the principles of causal conjecture and with causal models. Before then, there is a large amount of detailed ground to be covered, in developing the necessary language, concepts, and mathematical results. Given the ambitious objectives, it is not surprising that there is a new vocabulary to learn. In defining 'Humean events/variables' and 'Moivrean events/variables', for example, shafer distinguishes two different senses in which we talk of events and variables in a probability tree. Two of his fundamental new concepts are those of 'tracking' and 'sign'. He develops these over the middle portion of the book, to facilitate analysis of situations where unfolding circumstances change the probabilities of two or more events in an apparently related way, e.g., 'We say that a Moivrean event E is a positive sign of a Moivrean event G if the steps in nature's tree that change the probability of E also change the probability of G in the same direction'. The book brims with fascinating ideas and arguments, many of which attempt to build a more rigorous framework for philosophical ideas that have seen centuries of debate. The issues are of farreaching importance and considerable philosophical content. Much of the material in the book, though, is of a closely argued technical nature. Many pages are replete with definitions, propositions, and proofs. Though none of the mathematics is difficult, the book demands careful study. It is certainly not an easy read, despite its intriguing subjectmatter. One of the difficulties (at least for this reviewer) is setting aside the habits formed from many years' thinking within the sample space framework, and trying to see things from a naïve perspective. Shafer is well aware that his development of ideas from a new startingpoint may cause difficulties for those steeped in sample space traditions. He therefore addresses the relationships between the two lines of development at various junctures in the main text, and has included some supporting appendixes (together occupying over 100 pages) amplifying the comparisons between the two approaches. This is an erudite and scholarly work. It will probably have a rather mixed readership. On the one hand, it will be of great interest to those interested in probabilistic expert systems, particularly for the promise that it holds of assisting the development of the interpretational devices for those systems. On the other hand, it will hold fascinations for those concerned about the foundations and philosophy of probability and statistics. The book is handsomely produced, in an unusual squarish format, though the frequent probability trees illustrating the text do not quite match up to the standards of the typography as a whole. 

Other Reviews Review Essay by Vern W. Walker, Jurimetrics 39, No. 4, Summer 1999, pp. 391429. Review by Clark Glymour, Journal of the American Statistical Association, David J. Hand: Review of The Art of Causal Conjecture. AI Magazine 19(1): 131132 (1998). Advance praise from A. Philip Dawid, Professor of Statistics,
Advance praise from Judea Pearl, University of California, Los Angeles.
Advance praise from Joseph Y. Halpern, Computer Science Department, Cornell University.

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