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Cox's theorem
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==Cox's assumptions== Cox wanted his system to satisfy the following conditions: #Divisibility and comparability – The plausibility of a [[proposition]] is a real number and is dependent on information we have related to the proposition. #Common sense – Plausibilities should vary sensibly with the assessment of plausibilities in the model. #Consistency – If the plausibility of a proposition can be derived in many ways, all the results must be equal. The postulates as stated here are taken from Arnborg and Sjödin.<ref name="AS1999">Stefan Arnborg and Gunnar Sjödin, ''On the foundations of Bayesianism,'' Preprint: Nada, KTH (1999) — http://www.stats.org.uk/cox-theorems/ArnborgSjodin2001.pdf</ref><ref name="AS2000a">Stefan Arnborg and Gunnar Sjödin, ''A note on the foundations of Bayesianism,'' Preprint: Nada, KTH (2000a) — http://www.stats.org.uk/bayesian/ArnborgSjodin1999.pdf</ref><ref name="AS2000b">Stefan Arnborg and Gunnar Sjödin, "Bayes rules in finite models," in ''European Conference on Artificial Intelligence,'' Berlin, (2000b) — https://frontiersinai.com/ecai/ecai2000/pdf/p0571.pdf</ref> "[[Common sense]]" includes consistency with Aristotelian [[logic]] in the sense that logically equivalent propositions shall have the same plausibility. The postulates as originally stated by Cox were not mathematically rigorous (although more so than the informal description above), as noted by [[Joseph Halpern|Halpern]].<ref name="H99a">Joseph Y. Halpern, "A counterexample to theorems of Cox and Fine," ''Journal of AI research,'' 10, 67–85 (1999) — http://www.jair.org/media/536/live-536-2054-jair.ps.Z {{Webarchive|url=https://web.archive.org/web/20151125021821/http://www.jair.org/media/536/live-536-2054-jair.ps.Z |date=2015-11-25 }}</ref><ref name="H99b">Joseph Y. Halpern, "Technical Addendum, Cox's theorem Revisited," ''Journal of AI research,'' 11, 429–435 (1999) — http://www.jair.org/media/644/live-644-1840-jair.ps.Z {{Webarchive|url=https://web.archive.org/web/20151125022616/http://www.jair.org/media/644/live-644-1840-jair.ps.Z |date=2015-11-25 }}</ref> 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 notation: :The plausibility of a proposition <math>A</math> given some related information <math>X</math> is denoted by <math>A\mid X</math>. Cox's postulates and functional equations are: *The plausibility of the [[logical conjunction|conjunction]] <math>AB</math> of two propositions <math>A</math>, <math>B</math>, given some related information <math>X</math>, is determined by the plausibility of <math>A</math> given <math>X</math> and that of <math>B</math> given <math>AX</math>. :In form of a [[functional equation]] ::<math>AB\mid X=g(A\mid X,B\mid AX)</math> :Because of the associative nature of the conjunction in propositional logic, the consistency with logic gives a functional equation saying that the function <math>g</math> is an [[associativity|associative]] binary operation. *Additionally, Cox postulates the function <math>g</math> to be [[monotonic function|monotonic]]. :All strictly increasing associative binary operations on the real numbers are isomorphic to multiplication of numbers in a [[Interval (mathematics)|subinterval]] of {{closed-closed|0, +∞}}, which means that there is a monotonic function <math>w</math> mapping plausibilities to {{closed-closed|0, +∞}} such that ::<math>w(AB\mid X)=w(A\mid X)w(B\mid AX)</math> *In case <math>A</math> given <math>X</math> is certain, we have <math>AB\mid X=B\mid X</math> and <math>B\mid AX=B\mid X</math> due to the requirement of consistency. The general equation then leads to :<math>w(B\mid X)=w(A\mid X)w(B\mid X)</math> :This shall hold for any proposition <math>B</math>, which leads to ::<math>w(A\mid X)=1</math> *In case <math>A</math> given <math>X</math> is impossible, we have <math>AB\mid X=A\mid X</math> and <math>A\mid BX=A\mid X</math> due to the requirement of consistency. The general equation (with the A and B factors switched) then leads to :<math>w(A\mid X)=w(B\mid X)w(A\mid X)</math> :This shall hold for any proposition <math>B</math>, which, without loss of generality, leads to a solution ::<math>w(A\mid X)=0</math> ::Due to the requirement of monotonicity, this means that <math>w</math> maps plausibilities to interval {{closed-closed|0, 1}}. *The plausibility of a proposition determines the plausibility of the proposition's [[negation]]. :This postulates the existence of a function <math>f</math> such that ::<math>w(\text{not } A\mid X)=f(w(A\mid X))</math> :Because "a double negative is an affirmative", consistency with logic gives a functional equation ::<math>f(f(x))=x,</math> :saying that the function <math>f</math> is an [[Involution_(mathematics)#Involutions_in_mathematical_logic|involution]], i.e., it is its own inverse. *Furthermore, Cox postulates the function <math>f</math> to be monotonic. :The above functional equations and consistency with logic imply that ::<math>w(AB\mid X)=w(A\mid X)f(w(\text{not }B\mid AX))=w(A\mid X)f\left( {w(A\text{ not }B\mid X) \over w(A\mid X)} \right)</math> :Since <math>AB</math> is logically equivalent to <math>BA</math>, we also get ::<math>w(A\mid X)f\left( {w(A\text{ not }B\mid X) \over w(A\mid X)} \right)=w(B\mid X)f\left( {w(B\text{ not }A\mid X) \over w(B\mid X)} \right)</math> :If, in particular, <math>B=\text{ not }(AD)</math>, then also <math>A\text{ not } B = \text{not }B</math> and <math>B\text{ not }A=\text{ not }A</math> and we get ::<math>w(A\text{ not }B\mid X)=w(\text{not }B\mid X)=f(w(B\mid X))</math> :and ::<math>w(B\text{ not }A\mid X)=w(\text{not }A\mid X)=f(w(A\mid X))</math> :Abbreviating <math>w(A\mid X)=x</math> and <math>w(B\mid X)=y</math> we get the functional equation ::<math>x\,f\left({f(y) \over x}\right)=y\,f\left({f(x) \over y}\right)</math>
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