ramsey, f.p. - weight ot the value of knowledge, 1990

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The ritish Society for the Philosophy of Science

Weight or the Value of KnowledgeAuthor(s): F. P. RamseySource: The British Journal for the Philosophy of Science, Vol. 41, No. 1 (Mar., 1990), pp. 1-4Published by: Oxford University Press on behalf of The British Society for the Philosophy of

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Brit. J. Phil. Sci. 41 (1990) 1-4 Printed in Great Britain

Weightor the Value of

Knowledge'F. P. RAMSEY

PREAMBLE

FrankRamsey's ote 'Weight rthe ValueofKnowledge's a gem.In the note

Ramsey roveshatwhenknowledger nformations free tpays nexpectationo

acquiret. Collectingvidencemeans hat one is gamblingwithexpectation utthis is

alwayspreferableo not

gambling,n the

assumptionhatinformations

free(seeRamsay'sigurebelow).Ramseyalsoshowshow muchthe increase n

weights.ForaBayesian, utalso oranon-Bayesian,his sanimportantheoremsince t tellsus whywe shouldcontinue o collectnewevidence,.e.continue omake new observations. everalproofsof this theoremcan be foundin theliteraturelocus lassicuss I.J.Good's rticle On hePrinciplef TotalEvidence'[1967] andthe discussionn L.J.Savage's 1954] TheFoundationfStatistics;nearlynon-Bayesianiscussionftheprobleman befoundnC.S.Peirce's,Noteonthetheory ftheeconomy fresearch'1876) [1958];adiscussion fRamsey'snote canbefound n B.Skyrms 1990] andN.-E.Sahlin 1990]).

Forthose nterestedn

thehistoryofprobabilityheory t is a well-knownactthatRamseynhiscelebratedaperTruth ndProbability'1926)[1990]laid hefoundations f the modern heoryof subjectiveprobability.He showedhowpeople's eliefs nddesiresanbemeasuredyuseof abettingmethod nd hat f anumber f intuitive rinciplesf rationalbehaviour reaccepted, measureusedto representur'degrees fbelief'willsatisfy he laws ofprobability. e was thefirstone to prove he so-calledDutchbooktheorem.However,developinghetheoryof subjective robabilitye also laid the foundations f modernutilitytheoryand decision heory(i.e. twentyyearsbeforeJ. von Neumannand 0.Morgenstern1947] developedheirutility heorynTheoryfGamesndEconomic

Behaviorndalmosthirty earsbeforeL.J.Savage 1954] developedisBayesiandecision heory n TheFoundationsfStatistics). stoundinglynough, his noteshows hatRamsey lsohadaproof fthevalueofcollectingvidence,earsbeforetheworksofGood 1967],Savage 1954] andothers.

I have not alteredRamsey'sunorthodox otation orprobabilities,hichistakenfromJ.M.Keynes'A Treatise nProbability1921] (the word'weight'in thetitleofRamsey'snote alludesto Keynes'book).WhenRamsey,forexample,writes

aK/h=p, this should be read as the probabilityof a and K given h equals p. But

except forthis detail the note readseasily.NILS-ERICAHLIN

1 This note is part of the FrankRamsey Collection,document numbers 005-20-01 and 005-20-03, Archives of ScientificPhilosophy in the Twentieth Century,Hillman Library,University of

Pittsburgh. Quotedby permission of the University of Pittsburgh. All rights reserved.








2 F. P. Ramsey

Lemma.Supposea is an unknown proposition.4(p) is expectationofadvantagein regard to a if I expect it with probabilityp.

Then (p)> 0.

[Proof.]Suppose xi, x2, ... are variables I can alter by my action.

Goodto me if a=fl (xI, x2 ... )

a=f2 (X, X2.... )

If I expect a with prob[ability]p I act so as to maximize

pf + (1 -p)f2 for fixed given p

i.e. I make p--+ (1 -p) -2=0O r=1, 2 ... (1)ax, ax,

and pd2f1+(1 -p)d2f2<0 (2)

My expectation of advantage

O(p)= pfh + (1 -p)f2 with x's det[ermined]by (1)

I OfOOXrj dp

d'(up)= fl--f2 + p- + (l-p)r

= fi -f2 whatever p.

O (p)d

ffl,-f2

Now differentiating (1) w.r.t. p.

Of1Of2 02f, 02f2x0

s<p+ +(1-p) 0Ox, Ox, axraxs axrax, dp

df, df2 'af, Of2 dxr

Sdp p dax x2rdp

- xd a, + (1 p) af2 >o by 2).Sdpdp axraxs axrax,

r S

Valueof Knowledge

Suppose now I know h and expect a with prob[ability]p.

Expectationof advantage = 4(p).








Weightor the Valueof Knowledge 3

How muchis it worthwhile ofind out K?

where say K/h= A say y= 1 -

a a-= r, --- = SKh Kh

a aK aK

P.-=

+-= r+sp.h h h

Expectationof advantage if I find out K

= A4(r) + w4(s)

gain = PQ must be [positive] (unless p = r = s, K irrelevant)

r p s

expression for gain

=A).(p+r-p)+p (p+s-p)- 4(p) 2+p/=1, r)l+pjs=p

= (A + p - 1) (p) + {A(r-p) + p(s-p)} I'(p)

,(r-p)2

(s-p)22 2

= {A(r-p)2 + y (S-p)2}4,,(5).

Increase in weight say

= A(r-p)2 +

M(s-p)2= A2(r - S)2 +.22(r-

S)2 = Ai(rT- S)2.








4 F. P. Ramsey

REFERENCES

GOOD,.J. [1967]: 'On the principleof total evidence', BritishJournalfor hePhilosophy f

Science,17, pp. 319-321.

KEYNES,. M. [1921]: A Treatiseon Probability.Macmillan, London.

PEIRCE,. S. 'Note on the theory of the economy of research' (1876), in Collected apers

ofCharlesSandersPeirce, n A. W. Burks(ed.)HarvardUniversityPress,Cambridge,Mass. Volume 7, 7.140-7.161.

RAMSEY,. P. 'Truth and Probability' (1926) in his PhilosophicalPapers (Ed. D. H.

Mellor) [1990]. CambridgeUniversity Press, Cambridgeand New York,pp. 52-

94.

SAHLIN, .-E. [1990]: The Philosophyof F. P. Ramsey. Cambridge University Press,

forthcoming, Cambridge,Chapter 1.

SAVAGE, L.J. [1954]: The Foundations f Statistics.Wiley, New York.

SKYRMS,. [1990]: TheDynamicsof Rational Deliberation.Harvard University Press,

forthcoming, Chapter4.

VONNEUMANN, .and MORGENSTERN,. [1947]: Theory of Games and Economic Behavior,

2nd edition. Princeton University Press, Princeton.






ramsey, f.p. - weight ot the value of knowledge, 1990

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