Cox

Cox

[koks]
Cox, David, 1783-1859, English landscape painter, a follower of John Constable. He is best known for his watercolors of Welsh scenery, of which he produced a great number. Cox is well represented in the British and the Victoria and Albert museums and in the Birmingham Art Gallery.

See biographies by N. N. Solly (1875) and W. Hall (1881); study by F. G. Roe (1946).

Cox, George Barnsdale, 1853-1916, American politician, Republican boss of Cincinnati, b. Cincinnati. A former bootblack, he turned to real estate and local politics after selling his saloon in 1881. Cox acquired political control of the city in 1886 and held power till 1911, except for defeats in 1897 and 1905. His machine, in other hands, won again in 1913.
Cox, Jacob Dolson, 1828-1900, Union general in the Civil War and American statesman, b. Montreal, of a New York City family. Admitted to the Ohio bar in 1853, he was active in organizing the new Republican party there and served (1859-61) in the state senate. Cox, made a brigadier general of volunteers early in the Civil War, served ably in the Kanawha valley and Antietam campaigns and commanded in West Virginia (1862-63) and Ohio (Apr.-Dec., 1863). He later led a corps in the Atlanta campaign (1864), fought at Nashville (Dec., 1864), and finished his service with Sherman in North Carolina. He had risen to be a major general of volunteers and, returning home a hero, was elected governor of Ohio for the term 1866-68. Because he supported President Andrew Johnson on Reconstruction against the radical Republicans, he was not renominated. Nevertheless U. S. Grant, on assuming the presidency, made Cox his Secretary of the Interior. This was one of Grant's few good appointments. Cox, however, advocated and practiced civil service reform and opposed the President on other points, notably the move to annex Santo Domingo. The Republican spoilsmen had long been hostile to him, and in Oct., 1870, Cox resigned from the cabinet and became identified with the Liberal Republicans. He later served one term in Congress (1877-79), was dean of the Cincinnati Law School for 16 years beginning in 1881, and also served as president of the Univ. of Cincinnati from 1885 to 1889. He wrote ably on military affairs. His books include Atlanta (1882), The Battle of Franklin (1897), The March to the Sea (1898), and Military Reminiscences of the Civil War (1900). Kenyon Cox was his son.
Cox, James Middleton, 1870-1957, American political leader and journalist, b. Butler co., Ohio. After serving on the editorial staff of the Cincinnati Enquirer, he bought the Dayton (Ohio) Daily News (1898) and subsequently acquired several other papers in different states. He served in the U.S. House of Representatives (1909-13). As governor of Ohio (1913-15, 1917-21) he became prominent as a supporter of President Wilson. Nominated in 1920 as presidential candidate by the Democratic party with Franklin Delano Roosevelt as his running mate, Cox, a staunch supporter of the League of Nations, was soundly defeated by Warren G. Harding.

See his autobiography, Journey through My Years (1946).

Cox, Kenyon, 1856-1919, American painter, draftsman, and art critic, b. Warren, Ohio. He studied in Cincinnati, at the Pennsylvania Academy of the Fine Arts, and with Carolus-Duran and Gérôme in Paris. He worked in New York City, where he became an influential teacher at the Art Students League and the National Academy of Design. His portraits, figure pieces, and murals are academic in style. He painted murals for the Library of Congress, the state capitols of Iowa and Minnesota, and the public library of Winona, Minn. His portrait of Saint-Gaudens is in the Metropolitan Museum. Cox's writings on art include Old Masters and New (1905), The Classic Point of View (1911), and Concerning Painting (1917).

See his letters, ed. by H. W. Morgan (1986).

Cox, Louise Howland King, 1865-1945, American painter, b. San Francisco. She studied at the National Academy of Design and at the Art Students League, New York, under Kenyon Cox. whom she married in 1892. She is best known for her portraits of children and for her decorative paintings. May Flowers is in the National Gallery of Art, Washington, D.C.
Cox, Palmer, 1840-1924, American author and artist, b. Granby, Que. He is famous as the writer and illustrator of the Brownie stories for children (13 vol., 1887-1925).
Cox, Ross, 1793-1853, American fur trader. He joined John Jacob Astor's Pacific Fur Company in 1811 and went to the Astoria post on the supply ship Beaver. He was active in the trade in the Columbia river valley, entering the employ of the North West Company after Astoria was sold to that firm. His Adventures on the Columbia River (1831) is an entertaining as well as valuable historical source.
Cox, Samuel Sullivan, 1824-89, American statesman and legislator, b. Zanesville, Ohio. He traveled widely, practiced law, and was a newspaper editor before serving (1857-65) as a Congressman from Ohio. He moved (1865) to New York City and served again (1869-85) in the U.S. Congress. Cox argued for reforms in the civil service, worked to extend the scope of the census, and championed legislation for the development of the West. After serving (1885-86) as minister to Turkey, he again entered (1886) Congress. Among his books are A Buckeye Abroad (1852), Puritanism in Politics (1863), and Three Decades of Federal Legislation (1885).

See biography by D. Lindsey (1959).

Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability. As the laws of probability derived by Cox's theorem are applicable to any propositions, logical probability is a variety of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

Cox's assumptions

Cox wanted his system to satisfy the following conditions:

  1. Divisibility and comparability - The plausibility of a statement is a real number and is dependent on information we have related to the statement.
  2. Common sense - Plausibilities should vary sensibly with the assessment of plausibilities in the model.
  3. Consistency - If the plausibility of a statement can be derived in many ways, all the results must be equal.

The postulates as stated here are taken from Arnborg and Sjödin (1999). "Common sense" includes consistency with Aristotelian logic when statements are completely plausible or implausible.

The postulates as originally stated by Cox were not mathematically rigorous (although better than the informal description above), e.g., as noted by Halpern (1999a, 1999b). However it appears to be possible to augment them with various mathematical assumptions made either implicitly or explicitly by Cox to produce a valid proof.

Cox's axioms and functional equations are:

  • The plausibility of a proposition determines the plausibility of the proposition's negation; either decreases as the other increases. Because "a double negative is an affirmative", this becomes a functional equation

f(f(x))=x,,

saying that the function f that maps the probability of a proposition to the probability of the proposition's negation is an involution, i.e., it is its own inverse.

  • The plausibility of the conjunction [A & B] of two propositions A, B, depends only on the plausibility of B and that of A given that B is true. (From this Cox eventually infers that multiplication of probabilities is associative, and then that it may as well be ordinary multiplication of real numbers.) Because of the associative nature of the "and" operation in propositional logic, this becomes a functional equation saying that the function g such that

P(A mbox{and} B)=g(P(A),P(B|A))

is an associative binary operation. All strictly increasing associative binary operations on the real numbers are isomorphic to multiplication of numbers in the interval [0, 1]. This function therefore may be taken to be multiplication.

  • Suppose [A & B] is equivalent to [C & D]. If we acquire new information A and then acquire further new information B, and update all probabilities each time, the updated probabilities will be the same as if we had first acquired new information C and then acquired further new information D. In view of the fact that multiplication of probabilities can be taken to be ordinary multiplication of real numbers, this becomes a functional equation

y,fleft({f(z) over y}right)=z,fleft({f(y) over z}right)

where f is as above.

Cox's theorem implies that any plausibility model that meets the postulates is equivalent to the subjective probability model, i.e., can be converted to the probability model by rescaling.

Implications of Cox's postulates

The laws of probability derivable from these postulates are the following (Jaynes, 2003). Here w(A|B) is the "plausibility" of the proposition A given B, and m is some positive number.

  1. Certainty is represented by w(A|B) = 1.
  2. wm(A|B) + wm(AC|B) = 1
  3. w(A, B|C) = w(A|C) w(B|A, C) = w(B|C) w(A|B, C).

It is important to note that the postulates imply only these general properties. These are equivalent to the usual laws of probability assuming some conventions, namely that the scale of measurement is from zero to one, and the plausibility function, conventionally denoted P or Pr, is equal to wm. (We could have equivalently chosen to measure probabilities from one to infinity, with infinity representing certain falsehood.) With these conventions, we obtain the laws of probability in a more familiar form:

  1. Certain truth is represented by Pr(A|B) = 1, and certain falsehood by Pr(A|B) = 0.
  2. Pr(A|B) + Pr(AC|B) = 1
  3. Pr(A, B|C) = Pr(A|C) Pr(B|A, C) = Pr(B|C) Pr(A|B, C).

Rule 2 is a rule for negation, and rule 3 is a rule for conjunction. Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone (the conjunctive normal form), we can now handle any compound proposition.

The laws thus derived yield finite additivity of probability, but not countable additivity. The measure-theoretic formulation of Kolmogorov assumes that a probability measure is countably additive. This slightly stronger condition is necessary for the proof of certain theorems, however, it is not clear what difference countable additivity makes in practice.

Interpretation and further discussion

Cox's theorem has come to be used as one of the justifications for the use of Bayesian probability theory. For example, in Jaynes (2003) it is discussed in detail in chapters 1 and 2 and is a cornerstone for the rest of the book. Probability is interpreted as a formal system of logic, the natural extension of Aristotelian logic (in which every statement is either true or false) into the realm of reasoning in the presence of uncertainty.

It has been debated to what degree the theorem excludes alternative models for reasoning about uncertainty. For example, if certain "unintuitive" mathematical assumptions were dropped then alternatives could be devised, e.g., an example provided by Halpern (1999a). However Arnborg and Sjödin (1999, 2000a, 2000b) suggest additional "common sense" postulates, which would allow the assumptions to be relaxed in some cases while still ruling out the Halpern example.

The original formulation of Cox's theorem is in Cox (1946), which is extended with additional results and more discussion in Cox (1961). Jaynes (2003) cites Abel (1826) as first known instance of the associativity functional equation which is used in the proof of the theorem. Aczél (1966) refers to the "associativity equation" and lists 98 references to works that discuss it or use it, and gives a proof that doesn't require differentiability (pages 256-267).

See also

References and external links

  1. Niels Henrik Abel "Untersuchung der Functionen zweier unabhängig veränderlichen Gröszen x und y, wie f(x, y), welche die Eigenschaft haben, dasz f[z, f(x,y)] eine symmetrische Function von z, x und y ist.", Jour. Reine u. angew. Math. (Crelle's Jour.), 1, 11–15, (1826).
  2. R. T. Cox, "Probability, Frequency, and Reasonable Expectation," Am. Jour. Phys., 14, 1–13, (1946).
  3. R. T. Cox, The Algebra of Probable Inference, Johns Hopkins University Press, Baltimore, MD, (1961).
  4. Janos Aczél, Lectures on Functional Equations and their Applications, Academic Press, New York, (1966).
  5. Terrence L. Fine, Theories of Probability; An examination of foundations, Academic Press, New York, (1973).
  6. Edwin Thompson Jaynes, Probability Theory: The Logic of Science, Cambridge University Press (2003). — preprint version (1996) at http://omega.albany.edu:8008/JaynesBook.html; Chapters 1 to 3 of published version at http://bayes.wustl.edu/etj/prob/book.pdf
  7. Joseph Y. Halpern, "A counterexample to theorems of Cox and Fine," Journal of AI research, 10, 67–85 (1999) — http://www.cs.washington.edu/research/jair/abstracts/halpern99a.html
  8. Joseph Y. Halpern, "Technical Addendum, Cox's theorem Revisited," Journal of AI research, 11, 429–435 (1999) — http://www.cs.washington.edu/research/jair/abstracts/halpern99b.html
  9. Stefan Arnborg and Gunnar Sjödin, On the foundations of Bayesianism, Preprint: Nada, KTH (1999) — ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/06arnborg.ps — ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/06arnborg.pdf
  10. Stefan Arnborg and Gunnar Sjödin, A note on the foundations of Bayesianism, Preprint: Nada, KTH (2000a) — ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobshle.ps — ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobshle.pdf
  11. Stefan Arnborg and Gunnar Sjödin, "Bayes rules in finite models," in European Conference on Artificial Intelligence, Berlin, (2000b) — ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobc1.ps — ftp://ftp.nada.kth.se/pub/documents/Theory/Stefan-Arnborg/fobc1.pdf
  12. Michael Hardy, "Scaled Boolean algebras", Advances in Applied Mathematics, August 2002, pages 243–292 (or preprint); Hardy has said, "I assert there that I think Cox's assumptions are too strong, although I don't really say why. I do say what I would replace them with." (The quote is from a Wikipedia discussion page, not from the article.)
  13. Kevin S. Van Horn, "Constructing a logic of plausible inference: a guide to Cox’s theorem", International Journal of Approximate Reasoning, Volume 34, Issue 1, September 2003, Pages 3–24. (Or through Citeseer page)

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