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Chapter 2: The quantitative rules

  • p. 31, line after eqn. (2.44): It appears that the reference to (2.25) should be (2.40).
  • p. 31, eqn. (2.45): The constraint $0\leq x\leq 1$ should be $0 < x
\leq 1$, to avoid division by 0.
  • p. 32, eqn. (2.49): A right parenthesis is missing in the expression `` $S(1 - \exp{-q}$''.

  • p. 33, sentence starting at top of page: to account for the restriction $x>0$, this sentence should probably be rewritten as ``Using continuity, the only solution of this satisfying $\lim_{x \rightarrow 0} S(x) = 1$ is...''

  • p. 33, second paragraph: ``Again, Aczel (1966) derives the same result without assuming differentiability.'' (This refers to equation (2.58).) I checked out the Aczél reference in preparing a review paper on Cox's Theorem, and nowhere did I find anything like the result of equation (2.58); I can only assume that the result appears in some other work of Aczél. I did find, however, that a similar result appears in Paris's book, The Uncertain Reasoner's Companion.
  • p. 34, eqn. (2.65), second term after first "=" sign: $\overline{AB}$ should be $\overline{A}\;\overline{B}$.
  • p. 34, Exercise 2.2: In two places (lines 2-3 and equation (2.67)) a right parenthesis is missing:

    \begin{displaymath}p(C \mid (A_1 + A_2 + ... + A_n X) \end{displaymath}

    should be

    \begin{displaymath}p(C \mid (A_1 + A_2 + ... + A_n) X). \end{displaymath}

  • p. 40, second paragraph: ``The argument we have just given is the first `baby' version of the group invariance principle for assigning plausibilities; it will be extended greatly in Chapter 6, when we consider the general problem of assigning `noninformative priors.' '' I believe that ``Chapter 6'' should actually be ``Chapter 12'' (``Ignorance priors and transformation groups'').

  • p. 42, second full paragraph, line six: ``kelvin'' should be ``Kelvin.''

Commentary

Several people have written asking how to derive (2.50) from (2.48); here is the derivation:

  1. Rewrite (2.48) as $y = S(x) / (1 - \exp(-q))$.
  2. Note that, for all $z$, $1/(1-z) = 1 + z + O(z^2)$.
  3. Rewrite step 1 as

    \begin{displaymath}y = S(x) (1 + \exp(-q) + O(\exp(-2q))). \end{displaymath}

  4. Apply $S()$ to both sides:

    \begin{displaymath}S(y) = S[S(x) + S(x)\exp(-q) + S(x) O(\exp(-2q))]. \end{displaymath}

  5. Now do a Taylor series expansion around $y = S(x)$:

    \begin{eqnarray*}
S(y) &=&
S(S(x)) + S(x)\exp(-q) S'(S(x)) + S(x) O(\exp(-2q))...
...S(x)) \\
&=&
S(S(x)) + \exp(-q) S(x) S'(S(x)) + O(\exp(-2q))
\end{eqnarray*}

    since $S(x) S'(S(x))$ has no dependence on $q$.


Commentary: Additional treatments of Cox's Theorem

The following references not appearing Jaynes' book provide additional perspective on Cox's Theorem:

  • J. B. Paris, The Uncertain Reasoner's Companion: A Mathematical Perspective, Cambridge University Press, 1994. Chapter 3 proves a version of Cox's Theorem, with great care taken to explicitly list all of the assumptions required.

  • K. S. Van Horn, ``Constructing a logic of plausible inference: a guide to Cox's Theorem,'' International Journal of Approximate Reasoning 34, no. 1 (Sept. 2003), pp. 3-24. Reviews Cox's Theorem, explicitly listing the assumptions required and discussing (1) the intuition and reasoning behind these requirements, and (2) the most important objections to these requirements. (Preprint: Postscript, PDF.)

    • G. Shafer, ``Comments on `Constructing a logic of plausible inference: a guide to Cox's Theorem', by Kevin S. Van Horn,'' IJAR 35, no. 1 (Jan. 2004), pp. 97-105. Comments by Glenn Shafer (of Dempster-Shafer belief function theory). (Preprint: PDF, or try Glenn Shafer c.v. and look under ``Other Contributions.'')
    • K. S. Van Horn, ``Response to Shafer's comments,'' IJAR 35, no. 1 (Jan. 2004), pp. 107-110. Van Horn's rejoinder. (Preprint: Postscript, PDF.)

  • J. M. Garrett, ``Whence the laws of probability?'', in G. J. Erickson, J. T. Rychert, and C. R. Smith (eds.), Maximum Entropy and Bayesian Methods. Boise, Idaho, USA, 1997, Kluwer Academic Publishers. Another derivation of the laws of probability theory, very similar to the Cox derivation but starting from a single logical operation (NAND) instead of two (AND, NOT). (Online, slightly updated: Postscript, PDF)

Note that, like Cox, Jaynes does not explicitly list all of his precise assumptions in one place (although he gives desiderata that motivate them) -- Paris and Van Horn both address this issue.

Commentary: Consistency of Cox's axioms

Section 2.6.2 discusses the question of whether the rules of probability theory are consistent. Jaynes brings up Godel's result that no mathematical system can provide a proof of its own consistency, and later writes that ``These considerations seem to open up the possibility that, by going into a wider field by invoking principles external to probability theory, one might be able to prove the consistency of our rules. At the moment, this appears to us to be an open question.''

Actually, it is not an open question: the rules of probability theory can easily be proven consistent, and the proof can be found in any undergraduate mathematical text discussing set-theoretic probability theory. As I wrote in the above-mentioned review of Cox's Theorem,

But how do we know that our requirements [Cox's axioms] are not contradictory? How do we know that there is any system of plausible reasoning... that satisfies all of our requirements? The set-theoretical approach to probability theory may be taken as an existence proof that our requirements are not contradictory, by taking states of information to be [set-theoretical] probability distributions, and defining [state of information] $A,X$ to be the probability distribution obtained from $X$ by conditioning on the set of values for which $A$ is true. In the terminology of mathematical logic, set-theoretical probability theory then becomes the model theory for our logic, a tool to enable us to construct consistent sets of axioms (plausibility assignments from which we derive other plausibility assignments).

Viewing probability theory as an extension of the propositional calculus, Jaynes's ``wider field'' is just the predicate calculus with the axioms of finite set theory and real numbers added.

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