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Probability Theory: The Logic of Science
Books: Text Books: Construction: Item 3
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Probability Theory: The Logic of Scienceby E. T. Jaynes and G. Larry Bretthorst
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73 of 91 people found the following review helpful: Brilliant but attended by many misunderstandings, June 25, 2003 Reviewer:Michael Hardy (Minneapolis, MN, USA, for the Time Being) - To "pure" mathematicians, probability theory is measure theory in spaces of measure 1. To the extent to which you remain a "pure" mathematician, this book will be incomprehensible to you. To frequentist statisticians, probability theory is the study of relative frequencies or of proportions of a population; those are "probabilities". To Bayesian statisticians, probability theory is the study of degrees of belief. Bayesians may assign probability 1/2 to the proposition that there was life on Mars a billion years ago; frequentists will not do that because they cannot say that there was life on Mars a billion years ago in precisely half of all cases -- there are no such "cases". To _subjective_ Bayesians, probability theory is about subjective degrees of belief. A subjective degree of belief is merely how sure you happen to be. "Noninformative" _objective_ Bayesians assign "noninformative" probability distributions when they deal with uncertain propositions or uncertain quantities, and replace them with "informative" distributions only when they update them because of "data". "Data", in this sense, consists of the outcomes of random experiments. "Informative" _objective_ Bayesians -- a rare species -- ask what degree of belief in an uncertain proposition is logically necessitated by whatever information one has, and they don't necessarily require that information to consist of outcomes of random experiments. Jaynes is an "informative" objective Bayesian. This book is his defense of that position and his account of how it is to be used. "Pure" mathematicians will not find that this book resembles that branch of "pure" mathematics that they call probability theory. Jaynes rails against those he disagrees with at great length. Often he is right. But often he simply misunderstands them. For example, writing in the 1990s, he said that pure mathematicians reject the use of Dirac's delta function and its derivatives, and related topics. That is nonsense; the delta function has long been considered highly respectable, and required material in the graduate curriculum. Unfortunately Jaynes's misunderstandings may cause some others to misunderstand him when he is right. Statisticians are more informed than "pure" mathematicians and will disagree with Jaynes for better reasons. _Some_ statisticians will agree with him. Jaynes has many flaws, made all the more annoying by the fact that we need to overlook them in order to understand him. His message is important.
Product Review
'This is not an ordinary text. It is an unabashed, hard sell of the Bayesian approach to statistics. It is wonderfully down to earth, with hundreds of telling examples. Everyone who is interested in the problems or applications of statistics should have a serious look.' SIAM News
'This book could be of interest to scientists working in areas where inference of incomplete information should be made.' Zentralblatt MATH
' the author thinks for himself and writes in a lively way about all sorts of things. It is worth dipping into it if only for vivid expressions of opinion. The annotated References and Bibliography are particularly good for this.' Notices of the American Mathematical Society
Book Description
Going beyond the conventional mathematics of probability theory, this study views the subject in a wider context. It discusses new results, along with applications of probability theory to a variety of problems. The book contains many exercises and is suitable for use as a textbook on graduate-level courses involving data analysis. Aimed at readers already familiar with applied mathematics at an advanced undergraduate level or higher, it is of interest to scientists concerned with inference from incomplete information.
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