This book by the distinguished philosopher of probability, Ian Hacking, is titled An Introduction to Probability and Inductive Logic. It would have given a better idea of the scope of the book if the words "and Statistical Inference" had been added to the title. The focus of the book is on showing how probability theory has two functions: it uses deductive logic to find the properties of samples based on assumptions about a population, and inductive logic to draw conclusions about populations based on the measured characteristics or properties of samples.
Hacking explores and explains this whole structure in a very readable exposition, giving appropriate examples at every stage. He also weaves into his whole discussion the differences between the frequency theory of probability, which regards probability as essentially long run frequencies, and the Bayesian theory which regards probability as fundamentally concerned with belief and the revision of belief in the light of evidence. As he points out, while there are dogmatists who believe that only one of these views is correct, it is more fruitful to take an eclectic view which regards these two views as each more appropriate in certain situations.
Hacking also goes on to discuss the famous (or infamous?) philosophical problem first raised by David Hume, about the lack of any fundamental justification for inductive inferences. His discussion of how the frequentists as well as the Bayesians in different ways make an end run round the problem is indeed thought-provoking. He explains that the frequentists in effect treat Hume's problem as not material to what they do, by saying that statistical inference is about inductive behavior which is not guaranteed to be correct, but only correct most of the time. On the other hand, Bayesians in a different way make Hume's problem irrelevant — by saying that statistical inference is about revision of initial beliefs in the light of evidence, and is not designed to produce certain knowledge. It is only designed to incorporate additional information in a consistent or coherent way with initial beliefs. Hacking also ties up these discussions with Popper's theory of probability as propensity, and with Peirce's views of probability as concerned with faith, and the views of Keynes and Ramsey about probability.
Given the time constraints on introductory courses in statistics for undergraduates, these issues are never highlighted in such courses, and the focus is on techniques used in statistical inference, rather than on the nature of probability and the philosophical issues connected with it. It would be unrealistic to expect that this remarkable book could be used even as supplementary reading in such courses. But for any graduate course in statistics, I would say that this book should be required reading — and it would be very enjoyable required reading because of the relaxed and lucid style in which the book is written.
Ramachandran Bharath (firstname.lastname@example.org) is Visiting Professor of Mathematics at Colby Collge.
Part I. Logic: 1. Logic; 2. What is inductive logic?; Part II. How to Calculate Probabilities: 3. The gambler’s fallacy; 4. Elementary probability; 5. Conditional probability; 6. Basic laws of probability; 7. Bayes’ rule; Part III. How to Combine Probabilities and Utilities: 8. Expected value; 9. Maximizing expected value; 10. Decision under uncertainty; Part IV. Kinds of Probability: 11. What do you mean?; 12. Theories about probability; Part V. Probability as a Measure of Belief: 13. Personal probabilities; 14. Coherence; 15. Learning from experience; Part VI. Probability as Frequency: 16. Stability; 17. Normal approximations; 18. Significance; 19. Confidence and inductive behaviour; Part VII. Probability Applied to Philosophy: 20. The philosophical problem of induction; 21. Learning from experience as an evasion of the problem; 22. Inductive behaviour as an evasion of the problem.