Probability Theory: The Logic of Science

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 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.

One of the most important works of the 20th century (or any century) in both philosophy and physics, Jaynes' work lays the foundation for the physical ontology and epistemology of science. This book is the completion of what amounted to a lifetime of effort on Jaynes' part, dating back to the "Mobil Lectures" where he first laid out this approach to knowledge. It follows the world of Richard Cox, who demonstrated that Bayesian probability theory naturally follows from three simple axioms that also serve to establish the connection between evidence and plausible belief.In my opinion, this book is a required read for anyone who wishes to understand precisely how the scientific worldview is, in a mathematically defensible sense, the best possible worldview, the one that lets us optimally use evidence to develop an interlocked Bayesian network of evidence supported beliefs that can change and evolve as the evidence is accumulated. It also shows the critical connections between physics and statistical mechanics and Shannon's theorem in computational information theory, laying the foundation for a fair bit of modern physics as it demonstrates that physical entropy and information entropy are very much one and the same thing, from a certain point of view.