Probability Theory: The Logic of Science

I have rarely learned so much from one book. This book is somewhat unusual among mathematical texts in that it is heavy on prose and (compared to other texts) light on equations. However, don't get the idea that it is any less rigorous! It simply focuses on precisely what most math books neglect: exhaustive explanation of the concepts...and to very good effect. Jaynes (and his editor) are possibly the most articulate writers of mathematics I've ever read. If you can read equations like English, you may not appreciate this. The rest of us will. Summarizing the content: The book very exhaustively demonstrates how Bayesian statistical approaches subsume rather than compete with "orthodox" (sampling theory-derived) statistics. Importantly, it begins by deriving the sum and product rules (which in other texts are typically presented as axioms) from "common sense" considerations. In other words, what is usually treated as "given" in other statistics texts is shown to, in fact, depend on even more fundamental (and, thus, indisputable) considerations of what constitutes rational plausible reasoning. This places the whole endeavor of statistics on firmer ground than any other text I've seen. The book is worth buying for the first few chapters alone, but it just gets better from there. Jaynes goes on to link Bayes rule to information-theoretic considerations and build up probability as an extended form of logic (as the title implies). In some cases this yields a new and deeper understanding of "orthodox statistical practice." In others it exposes (and explains) the absurdities of strictly frequentist approaches. Again, I have rarely learned so much from one book. One caveat: It does not at all require a statistics background, but, obviously, some of Jaynes (mildly polemical) discourse will, of course, be lost on you without it.

To me, this is the greatest book ever written on Statistics. I have studied statistics for the past 22 years and I have been teaching statistics for the past 10 years. I only got to know this book a couple of years ago. Many many conceptual issues that I have had in Statistics have been clarified from a careful study of this book. Jaynes had a deep understanding not only of Bayesian Statistics but also of Frequentist Statistics. Everything that he says about Frequentist "Orthodox" Statistics is correct (although often it took me many months to fully understand what he is saying). The ideas and messages of this book significantly differ from what is taught in pretty much all other statistics books. Here is one example, the Gaussian distribution is heavily used in statistical analysis. Most textbooks are pretty much apologetic about this overuse of the Gaussian distribution and struggle to suggest alternative methods. Jaynes, on the other hand, says (in Chapter 7) that the range of validity for the application of the Gaussian distribution in data analysis is actually "far wider that is usually supposed". A major highlight of the book is the focus on history. Very careful historical accounts are presented as to how the greats of the field (like Gauss, Laplace, Cox, Fisher etc) approached data analysis. This stuff again cannot be found in any other book in the field. I have been using this book heavily in pretty much anything I teach these days and, as a consequence, teaching statistics has been a much more pleasurable experience than before. Jaynes apparently originally wanted to write a sequel to this book focussing on more advanced applications. It is a pity that he passed away before he could write the sequel. I recommend readers to the outstanding books by MacKay and by von der Linden-Dose-von Toussaint for numerous interesting and nontrivial applications of Probability Theory (Bayesian Statistics) to Data Problems. I would also like to recommend (as sequels to reading Jaynes) the books of David Blower which clarify and complement the ideas of Jaynes. For readers interested in learning more about the various issues, pitfalls and shortcomings of Frequentist "Orthodox" statistics, I would like to recommend the collected works of Dev Basu.

I haven't finished reading this book yet, but the chapters I read so far gave me so much understanding of issues that are either obscure or absent in other probability and statistics books - but are of great practical importance - that I decided recommend it here. It is true Jaynes' style is caustic against positions that are contrary to his owns. But he is very convincing on the reasons he gives to pinpoint the big holes in the so called "orthodox" school of probability and statistics. Besides, the book is very lengthy, without being prolix, on its explanations, making it very pedagogical. Constrasting with that, nevertheless, Jaynes sometimes proposes examples that I believe only a mathematician or physicist with specific knowledge of the subject mentioned by the author will be able to follow. But those parts do not impact understanding of the main ideas. It must be noted also that "Probaility theory: the logic of science" is mainly a theory book. Its goal is to present probability as an extension of deductive logic. It only brings a small number of exercises. The best thing about this book, at least for me, is having a style that really makes me look forward reading the next page, something very rare for a technical book. In fact, the only other book I came across that had that virtue was the "Feynman Lectures on Physics".

Soy científico, y nunca había entrado en la visión frecuentista de la probabilidad. Me parecían un conjunto de recetas basadas en unas hipotésis cuestionables. Este libro explica la probabilidad como una extensión de la lógica aristotélica, cuando no hay certezas absolutas. Entra en cuestiones aparentemente filosóficas sobre que base tiene la inducción. Los argumentos son claros y simples. La mayoría de casos que trata son bastante sencillos, muy tratables analíticamente. Quizás debe ser complementado con algun tratado más aplicado, pero como base de la probabilidad Bayesiana, me parece fantástico.

Love the content of the book (5 stars for the content), but the delivery could have been handled nicely: the corners of the book were damaged. It seems that such cases happened more often in recent years.

The book is print perfect and I love how Jaynes described Probability theory as a decisive mechanism for a robot. After reading this book, you might also consider being a statistician of Bayesian school of thought.

Possibly my favorite textbook ever. Jaynes is lucid and direct. This textbook is almost a polemic. I was a stringent frequentist, but after reading this book, I'm a diehard Bayesian. Amazing. Jaynes is a brilliant mind. Just an amazing, entertaining book. This will be on my bookshelf for the next 40 years.

Considering this is a weighty book about the fundamentals and history of probability theory, it is actually quite entertaining with humour, stories, an engaging style and vitriolic personal criticism (generally justified) of the people who fought hard to defend their mistaken positions by dismissing the ideas that Jaynes promoted. It can be a little over-wordy, opinionated and pompous in places, and has small sections missing because he unfortunately died before it was completed. It is however an absolute gem that rewards re-reading over an extended period of time, and will make anyone who has to deal with measuring and reasoning about uncertain systems - i.e. all scientists, engineers & economists in my opinion - think differently about what can be done as objectively as possible and how they should extract the maximum amount of information from measured data and make optimal inferences. Modern Bayesian theory is becoming the basis for solving Inverse Problems so if you are in this area then have a look. His description of probability distributions as "carriers of uncertain information about unknowns" rather than the traditional and flawed classical view of "behaviour of selected summary statistics in the limit of an infinite amount of repeated random events" (whatever that means!) is an indicator of the different perspectives. Anyone who wants to understand what probability theory actually _is_ at a fundamental level and have their mind opened up to how they can apply it in their area should have a look and strap themselves in for the ride. Highly recommended. If you want a more compact and introductory book with an applied focus and examples then I strongly recommend Sivia & Skilling: Data Analysis: A Bayesian Tutorial