Algebra of Probable Inference

by Richard T. Cox (Author)

6 customer reviews

ISBN-13: 978-0801869822

ISBN-10: 080186982X

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In Algebra of Probable Inference, Richard T. Cox develops and demonstrates that probability theory is the only theory of inductive inference that abides by logical consistency. Cox does so through a functional derivation of probability theory as the unique extension of Boolean Algebra thereby establishing, for the first time, the legitimacy of probability theory as formalized by Laplace in the 18th century.

Perhaps the most significant consequence of Cox's work is that probability represents a subjective degree of plausible belief relative to a particular system but is a theory that applies universally and objectively across any system making inferences based on an incomplete state of knowledge. Cox goes well beyond this amazing conceptual advancement, however, and begins to formulate a theory of logical questions through his consideration of systems of assertions―a theory that he more fully developed some years later. Although Cox's contributions to probability are acknowledged and have recently gained worldwide recognition, the significance of his work regarding logical questions is virtually unknown. The contributions of Richard Cox to logic and inductive reasoning may eventually be seen to be the most significant since Aristotle.

Algebra of Probable Inference

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Editorial Reviews

Review

"[This book] is, in my opinion one of the most important ever written on the foundations of probability theory, and the greatest advance in the conceptual, as opposed to the purely mathematical, formulation of the theory since Laplace."

(E. T. Jaynes American Journal of Physics)

"Transformed my view of probability and enriched my career as a physicist."

(Bruce Partridge Physics Today)

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Product details

Paperback: 127 pages
Publisher: Johns Hopkins University Press (November 28, 2001)
Language: English
ISBN-10: 080186982X
ISBN-13: 978-0801869822
Product Dimensions: 6 x 0.3 x 9 inches
Shipping Weight: 8.8 ounces (View shipping rates and policies)
Average Customer Review: 6 customer reviews
Amazon Best Sellers Rank: #940,339 in Books (See Top 100 in Books)
- #142 in Books > Science & Math > Mathematics > Pure Mathematics > Algebra > Abstract
- #1277 in Books > Textbooks > Science & Mathematics > Mathematics > Algebra & Trigonometry
- #1651 in Books > Textbooks > Science & Mathematics > Mathematics > Statistics

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Interesting

April 25, 2014

Format: PaperbackVerified Purchase

... but not something to change my practice.

This book starts with basic notions of probability (conditional probability, entropy, etc.) and expands them in the direction of Boolean algebra. Although some notions seem to extend well to later developments in soft logics, this development stays very close to traditional probabilistic reasoning. The manipulations are clear, well justified, and meticulously developed. But, as a potential user, this struck me as filling in gaps that had never really bothered me rather than expanding the range of logical mechanism.

But, whether I put this to use or not, I find it interesting for at least two reasons. First, it predates many of the other soft logic developments by many years. Consciously or not, directly or not, later workers owe many of their more distant explorations to first steps like these. Second, it's simply a pleasure to read. I've found this true again and again: if you want to understand something abstruse, skip the "explainers" and go to the primary source. Goedel, Heisenberg, and many others stand behind Cox as founders in their fields, and stand unexcelled as proponents of their insights. Information, like water, becomes clearer as you go upstream. Cox seems to be the original spring from which many downstream developments arose. Even if his thoroughness might wear on a pragmatic reader, his development pulls you along. Rigorous proofs bring certainty, of a sort, but not always understanding. Given this book's limited and specific range, it offers real understanding.

-- wiredweird

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Christopher A. Meli

impressive and elegant

May 30, 2013

Format: Paperback

I decided to read this short book in preparation for tackling Probability Theory: The Logic of Science by Jaynes. Where I have the knowledge, I do concur with the other reviews - the content is simple in notation and wording but profound in content, taking a few easily-agreed axioms and using them to derive a number of central theorems of probability. From this perspective the book is singular, but I came away disappointed because few implications of this strengthening of the infrastructure were discussed. I guess I was looking for more "punch lines" like the restatement of Hume at the very end of the book; otherwise I came away with the feeling you might have if you spend a lot of money having the foundation of your house strengthened - what you thought was firm before is now truly firm, but otherwise you can't see much difference. I do agree with another reviewer that the book definitely deserves more than one pass, so I might come away with a better impression on second reading.

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John S. Ryan

Degrees of belief as an extension of Boolean logic

August 1, 2002

Format: PaperbackVerified Purchase

This is a great, great book that I'm absolutely ecstatic to see back in print. I was introduced to it when I was in graduate school (mathematics) and rooming in the house of a physics professor who swore by Richard Threlkeld Cox's account of subjective probability. I haven't had a copy of it in my hand for nearly twenty years; I happened across this page today and ordered it at once. So pardon me while I gush:
What Cox accomplishes in this deceptively slim volume is amazing. He places Bayesian probability theory on an axiomatic foundation, as a natural extension of Boolean logic, identifying probabilities with degrees of subjective belief in propositions rather than directly with frequencies of events (though he also argues that the subjectivist interpretation accords with the frequentist interpretation whenever the latter makes sense at all).
Essentially, he shows that the ordinary laws of probability theory are normative laws of thought that apply to degrees of belief in propositions, and that we have to conform to them if we want to think consistently.
If you like math and logic books, you'll find this one eminently readable; I haven't seen it in years and yet I still remember the stunning clarity of Cox's rigorous exposition.
This is the book that originally sold me on Bayesianism. If you have any interest in this subject at all, grab this one while it's available.

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Michael Hardy

The best introduction to logical probability theory

February 6, 2002

Format: PaperbackVerified Purchase

That Gian-Carlo Rota and I both admired this book largely
explains why I have my present position at MIT. But I
cannot write book reviews the way Rota did.
Why should the conventional sum and product rules of
probability hold when probabilities are assigned, not
to *events* that are *random* according to their
relative frequencies of occurrence, nor to subsets of
populations as proportions of the whole, but rather
to *propositions* that are *uncertain* according to the
degree to which the evidence supports them? The tenet
that the same rules should apply to such "degrees of
belief," whether they are "subjective" probabilities or
"logical" probabilities, is the essence of Bayesianism.
The relative merits of Bayesian and frequentist methods of
statistical inference have been debated for decades. But
seldom is the question with which I started this paragraph
addressed. Several answers to that question have been
proposed. Richard Cox's book embodies one of them.
Many writers on foundations of statistical inference are
callously imprecise about the kind of topic dealt with
in this book. Cox is their antipode, writing not only
clearly, but supremely efficiently, beautifully, perhaps
sometimes poetically, about functional differential
equations and about delicate philosophical questions.
Cox also deals with the relationship between entropy and
distributive lattices. Shannon entropy is to distributive
lattices as probability is to Boolean algebras. I do not
think Cox was familiar with standard work on lattice theory.
He never uses the word "lattice," nor other standard
lattice-theory nomenclature.

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Most recent customer reviews

Cox understood Keynes better than Ramsey and de Finetti but

Cox understood Keynes's logical approach to probability better than either Ramsey or de Finetti but ultimately fails to improve substantially on Keynes's work due to his...Read more

Published on July 27, 2004 by Michael Emmett Brady

Like a ten-pound textbook, in only 130 pages

This book contains the fundamental argument justifying Laplace's original theory of probability. Laplace justified his theory by basic intuitive considerations, which left it open...Read more

Published on June 20, 2002 by Corey Yanofsky

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