the philosophy of statistics

8/13/2019 The Philosophy of Statistics

1/46

The Philosophy of Statistics

Author(s): Dennis V. LindleySource: Journal of the Royal Statistical Society. Series D (The Statistician), Vol. 49, No. 3(2000), pp. 293-337Published by: Blackwell Publishingfor the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2681060.

Accessed: 13/02/2011 20:00

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at.http://www.jstor.org/action/showPublisher?publisherCode=black..

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

Blackwell PublishingandRoyal Statistical Societyare collaborating with JSTOR to digitize, preserve and

extend access toJournal of the Royal Statistical Society. Series D (The Statistician).

http://www.jstor.org
http://www.jstor.org/action/showPublisher?publisherCode=blackhttp://www.jstor.org/action/showPublisher?publisherCode=rsshttp://www.jstor.org/stable/2681060?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=blackhttp://www.jstor.org/action/showPublisher?publisherCode=blackhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2681060?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=rsshttp://www.jstor.org/action/showPublisher?publisherCode=black


2/46

The Statistician2000)49, Part3, pp. 293-337

The philosophy of statisticsDennis V. LindleyMinehead,UK[Received June1999]Summary.This paper puts forward n overallview of statistics. t s argued that tatistics s thestudyofuncertainty.he manydemonstrations hatuncertaintiesan only combine according othe rules of theprobabilityalculus are summarized.The conclusion s that tatistical nference sfirmlyased on probability lone. Progress is therefore ependent on the construction f aprobability odel; methods fordoing this are considered. It is argued thatthe probabilities repersonal.The roles of likelihood nd exchangeability re explained.Inference s onlyof value if tcan be used, so the extension o decisionanalysis, ncorporatingtility,s related o risk nd totheuse ofstatistics nscience and law.The paperhas been writtennthehopethat twillbe intelligibletoallwho are interestedn tatistics.Keywords: onglomerability;ata analysis;Decisionanalysis; Exchangeability; aw; Likelihood;Models;Personalprobability; isk;Scientificmethod;Utility

1. IntroductionInstead f discussing specific roblem, hispaper provides n overviewwithinwhichmoststatisticalssues an be considered.Philosophy'nthe itle s used nthe enseof Thestudy fthegeneral rinciples f some particular ranch fknowledge,xperienceractivity'Onions,1956).The word asrecentlycquired reputationor eing oncernedolelywith bstractssues,divorcedromeality. y ntentionere s toavoid xcessive bstractionnd o dealwith racticalmattersoncerned ith ur ubject.fthepractitionerhoreads his aperdoesnotfeel hat hestudy as benefitedhem,mywriting ill havefailednone of tsendeavours. hepaper ries odevelop wayof ookingt statisticshatwillhelpus,as statisticians,odevelop etter soundapproachoanyproblem hichwemightncounter.echnicalmattersave argelyeen voided,notbecausethey re not mportant,ut n the nterestsffocusingn a clearunderstandingfhow a statisticalituationanbe studied. t someplaces,mattersf detailhavebeen omittedohighlighthekey dea. Forexample, robabilityensities avebeen used without n explicitmentionfthedominatingeasure owhich hey efer.Thepaperbegins y recognizinghat tatisticalssuesconcern ncertainty,oing n toarguethatuncertaintyan onlybe measured y probability.his conclusion nables a systematicaccount f inference,asedon theprobabilityalculus, o be developed, hich s shown o bedifferentromome conventionalccounts. he likelihood rincipleollows rom hebasic roleplayed y probability.herole ofdataanalysisn the onstructionfprobability odels nd thenature f models re next iscussed. hedevelopmenteads to a method fmaking ecisionsndthe nature f risk s considered. cientificmethodnd itsapplicationo some egal issues areexplainedwithinheprobabilisticramework.heconclusions thatwehave here satisfactorygeneraletof tatisticalrocedures hose mplementationhouldmprovetatisticalractice.

Addressor orrespondence:ennisV Lindley, Woodstock", uayLane,Minehead, omerset,A245QU,UK.E-mail: [email protected]? 2000 RoyalStatistical ociety 0039-0526/00/49293


3/46

294 D. V. LindleyThe philosophy erepresented laces more mphasis n modelconstructionhan n formalinference.n this tagreeswithmuch ecent pinion. reason or his hange f emphasiss thatformalnferences a systematicrocedure ithin he alculus fprobability. odel construction,

bycontrast,annot e so systematic.The paper roseoutofmy xperiencest theSixthValencia onferencen Bayesian tatistics,held nJune 998 Bemardo tal., 1999). Although was impressed y theoverall uality fthepapers nd the ubstantialdvancesmade,many articipantsid not eemtomefullyoappre-ciate heBayesian hilosophy.hispaper s an attempto describemyversion f that hilosophy.It is a reflectionf 50 years' tatistical xperience nd a personal hangefrom frequentist,throughbjective ayes, o the subjective ttitude resented ere.No attempt as been made toanalysendetail lternativehilosophies,nly o ndicate hereheir onclusionsifferrom hosedeveloped ere ndto contrasthe esultingracticalmethods.

2. StatisticsTo discussthephilosophyf statistics,t is necessary o be reasonably lear what t is thephilosophyf,not nthe enseof a precise efinition,o that his s 'in', that s 'out',butmerelytobe able to perceive tsoutlines. he suggestion ere s that tatisticss the tudy f uncertainty(Savage, 1977): that tatisticiansreexpertsnhandling ncertainty.heyhavedeveloped ools,like standardrrors nd significanceevels, hatmeasure heuncertaintieshatwe might eason-ablyfeel.A check f howwellthis escriptionf our ubject greeswithwhatweactually ocanbeperformedy ookingt thefoureries f ournals ublished y heRoyalStatisticalocietyn1997. These embrace ssues as diverse s social accountingnd stable aws. Apart rom fewexceptions,ikethe lgorithmsection whichhassubsequentlyeen abandoned) nd a paper neducation,ll thepapersdeal either irectly ithuncertaintyr withfeatures,ike stable aws,which rise nproblemshat xhibit ncertainty.upport or hisview of our ubjects providedby the fact that statisticslaysa greater ole in topicsthathave variability,ivingrise touncertainty,s an essentialngredient,han nmoreprecise ubjects.Agriculture,or xample,enjoys close associationwith tatistics, hereas hysics oes not. Notice that t is onlythemanipulationf uncertaintyhat nterests s. We are notconcernedwiththematterhat suncertain. hus we do notstudyhe mechanismfrain;onlywhethertwill rain.Thisplacesstatisticsna curious ituationnthatweare, spractitioners,ependentnothers. heforecastfrainwillbe dependentn bothmeteorologistndstatistician.nly s theoreticiansanwe existalone.Even therewesufferfwe remain oo divorcedrom he cience.The termclient'will beused nreferenceo theperson, .g. scientistr awyer, ho encountersncertaintyn their ieldof tudy.Thephilosophical osition dopted ere s that tatisticss essentiallyhe tudy funcertaintyand that he statistician'sole is to assistworkersn other ields, heclients,who encounteruncertaintyn theirwork.npractice,here s a restrictionnthat tatisticss ordinarilyssociatedwith ata; nd t sthe ink etweenheuncertainty,rvariability,n thedata ndthatn the opicitself hathas occupied tatisticians.omewriters ven restricthedatato be frequency ata,capableofnear-identicalepetition. ncertainty,wayfrom ata,has rarely eenof statisticalinterest.tatisticianso nothavea monopolyf studies funcertainty.robabilists iscusshowrandomnessnonepart f a systemffects ther arts. hus themodelfor stochasticrocessprovides redictionsbout hedata hat heprocesswillprovide. hepassagefromrocess odatais clear; t swhenweattemptreversalndgofrom ata oprocess hat ifficultiesppear. hispaper smainly evoted othis astphase, ommonlyalled nference,ndthe ction hattmightgenerate.


4/46

Philosophyf tatistics 295Notice thatuncertaintys everywhere,ot ust in science or even in data. It providesmotivation or ome aspects of theologyBartholomew,988). Therefore,he recognitionfstatisticss uncertaintyould mply n extensive ole for tatisticians.fa philosophical osition

can be developed hatembraces ll uncertainty,t will provide n importantdvance n ourunderstandingftheworld.Atthemomenttwould e presumptiveo claim o much.3. UncertaintyAcceptance hat tatisticss the tudy f uncertaintymplies hatt s necessary o nvestigatehephenomenon. scientificpproachwouldmeanthemeasurementf uncertainty;or, o followKelvin, t is onlyby associating umbers ith ny scientificoncept hat he concept an beproperlynderstood.he reason ormeasurements not ustto makemore recise henotion hatwe are more ncertainbout he tock-markethan bout he unrising omorrow,ut o be ableto combine ncertainties.nlyexceptionallys there ne element f uncertaintyn a problem;more ealisticallyhere re several. n thecollection f data, here s uncertaintynthe amplingunit, ndthen nthenumber eportednthe ampling.n an archetypaltatisticalroblem,hereis uncertaintynbothdata and parameter.he central roblem s thereforehe combinationfuncertainties.ow therulesfor hecombinationf numbersre especially imple. urthermore,numbersombine n twoways, ddition nd multiplication,o leading o a richness f deas. Wewant o measure ncertaintiesn order o combine hem.A politician aid thathe preferredadverbs onumbers.nfortunatelyt s difficulto combine dverbs.How is thismeasuremento be achieved?All measurements based on a comparison ithstandard. or engthwe refer o theorange-redineof thekrypton-86sotope.Thekeyrole ofcomparisons eans hat here re noabsolutesntheworld fmeasurement.his s a pointwhichwe shall eturno nSection 1. t s thereforeecessaryo find standardor ncertainty.everalhave been suggested ut the simplest s historicallyhe first, amely ames of chance.Theseprovidedhefirst ncertaintieso be studied ystematically.etus thereforese as our tandardsimple ame.Consider efore ouan urn ontainingknown umber of balls that re as nearlydenticalas modernngineeringan makethem. uppose hat ne ball is drawn t random rom heurn.Forthis o makesense, t s needful o define andomness.magine hat heballs are numberedconsecutivelyrom toN andsuppose hat,t no cost oyou,youwere fferedprize fball 57were rawn. uppose, lternatively,hat ouwere fferedhe ameprize fball 12weredrawn.fyouare indifferentetween he twopropositionsnd, n extension,eween nytwo numbersbetween and N, then,foryou,theball is drawn t random.Notice that he definitionfrandomnesss subjective;tdepends nyou.What s random or nepersonmaynotbe randomfor nother. e shallreturno this spectnSection .Having aid what s meant ythedrawing fa ball at random, orgethenumbersndsupposethatR of the balls are red and the remainder hite, hecolouring otaffecting ouropinionof randomness. onsider he uncertain vent hat heball,withdrawnt random,s red. Thesuggestions that hisprovides standard oruncertaintynd that hemeasure s R/N, theproportionf red balls in the urn.There s nothing rofound ere,being ust a variant ftheassumptionn which amesof chance re based. Nowpasstoanyevent, rproposition,hichcan either appen rnot,be true r false. t is proposed o measure our ncertaintyssociatedwith heevent appening y comparison ith he tandard.fyouthinkhat heevent s ustasuncertains the andomrawingf a redballfromn urn ontaining balls,ofwhichR arered,then he vent as uncertainty/Nforyou.R andN are foryou o choose.ForgivenN, t s easyto see that here annot e more han ne suchR. There s now measure funcertaintyor ny


5/46

296 D. V.Lindleyevent rproposition.efore roceeding,etus consider hemeasurementrocess arefully.A serious ssumptionas beenmade hat he oncept funcertaintyan be isolated rom therfeatures.n discussing andomness,twas useful o compare prizegivenunder ifferentir-cumstances,all 57 or ball 12. Rewardswill notnecessarily ork n thecomparison f an eventwith standard. or example, uppose hat he eventwhoseuncertaintys being ssessed s theexplosion f a nuclearweapon,within 0 milesofyou,nextyear.Then a prizeof?10000, say,will be valueddifferentlyhen a red ball appearsfromwhen t will when fleeing rom heradiation.hemeasurementrocess ust described ssumes hatyou can isolate heuncertaintyofthenuclear xplosion romtsunpleasant onsequences. or themoment e shallmaketheassumption,eturningo it in Section 17 to showthat,n a sense, t is not mportant.amsey(1926), whoseworkwillbe discussed n Section , introducedhe oncept f an ethically eutralevent'forwhich hecomparison ith heurnpresentsewer ifficulties.uclearbombs renotethicallyeutral.Incontrast,otice n assumptionhat as notbeenmade.For ny vent,ncludinghenuclearbomb,thas notbeen assumed hat ou can dothemeasuremento determine (andN,but hatonlyreflectsheprecisionnyour ssessment ftheuncertainty).ather,we assumethatyouwouldwishto do it,wereyouto knowhow.All thats assumed f anymeasurementrocess sthatt s reasonable, ot hatt caneasilybe done.Because you do notknowhowto measure hedistance o ourmoon,tdoes notfollow hat oudo notbelieve nthe xistencef a distance o t.Scientists ave spentmuch effort n the accurate eterminationf length ecausetheywereconvincedhat heconcept f distancemade sense nterms f kryptonight. imilarly,t seemsreasonableoattempthemeasurementfuncertainty.4. Uncertaintynd probabilityIthas beennoted hat prime eason or hemeasurementfuncertaintiess to be able to combinethem,o letus seehowthemethoduggestedccomplisheshis nd. Suppose hat, ftheN ballsin theurn,R arered,B are blueandtheremainderhite. hen heuncertaintyhat s associatedwith hewithdrawalfa coloured all s (R + B)/N = R/N+ B/N,the umoftheuncertaintiesassociatedwith ed, nd with lue,balls.The sameresultwillobtain or ny wo xclusiveventswhose uncertaintiesre respectively/N and B/N and we have an addition ule foryouruncertaintiesfexclusive vents.Next, uppose gainthatR ballsare redandtheremaining - R white; t the ametime,arespottednd the emaining - S plain.The urn hen ontains ourypes fball,ofwhich netype s both pottedndred, f which henumbers sayT Thentheuncertaintyssociatedwiththewithdrawalf a spotted ed ball is T/N,which s equalto R/N X T/R,theproductf theuncertaintyfa redball and that fspottedallsamong hered.Again he ameresultwillapplyfor nytwoevents eing omparedwith oloured nd with potted allsand we havea productrulefor ncertainties.The additionndproductulesust obtained,ogetherith he onvexityule hat hemeasure-mentR/N always ies inthe convex)unitnterval,rethedefiningulesofprobability,t leastfor finiteumber feventssee Section ). The conclusionsthereforehat hemeasurementsfuncertaintyan be described ythecalculusofprobability.nthereverse irection,herulesofprobabilityeduce imply o the rulesgoverningroportions.ncidentally,hishelpsto explainwhy requentistrgumentsre often o useful: he ombinationf uncertaintiesan be studied yproportions,rfrequencies,na group, ere fballs.Themathematicalasis ofprobabilitysverysimple nd t sperhaps urprisinghat tyields omplicatednduseful esults.Theconclusions rethat tatisticiansreconcerned ith ncertaintyndthat ncertaintyan


6/46


7/46

298 D. V. Lindley5. ProbabilityThe conclusion s thatmeasurementsf uncertainty ustobey the rules of the probabilitycalculus.Other ules, ike thoseof fuzzy ogic or possibilityheory,ependentn maxima ndminima, ather han ums and products, re out. So are some rules used by statisticians;eeSection 6. All thesederivations,hether ased on balls in urns,gambles, coring ules ordecision-making,rebased on assumptions.ince hese ssumptionsmply uch mportantesults,it s proper hat hey reexamined ith reat are.Unfortunately,hegreatmajorityf tatisticiansdo notdo this. ome deny he entral esult bout robability,ithoutxploringhe easons or t.It s not hepurpose f this aper o provide rigorousccount ut etus lookat one assumptionbecauseof ts ater mplications. s with ll the ssumptions,t s intendedo be self-evidentndthat ouwouldfeelfoolishfyouwere o be caught iolatingt. t is based ona primitiveotionofone event eingmore ikely han nother.'Likely' s notused n ts echnicalense, ut s partof normal nglish sage.) We writeA is more ikely hanB' as A > B. The assumptions that fAl and A2 are exclusive, nd the same is trueof B, and B2, thenAi Bi (i = 1, 2) implyAl U A2 B, U B2. (Al U A2 means heevent hat s truewheneveritherAl orA2 s true.)Wemight eelunhappyfwe thoughthat henext erson o pass through doorwas more ikely o bea non-whiteemale, I,than non-white ale,B,, that white emale, 2, s more ikely hanwhitemale, B2, yeta female,AIU A2 was less likely han male,B, U B2.The developmentsoutlinedbove start rom ssumptions,r axioms, f this haracter.he importantoint s thatthey ll lead to probabilityeing he nly atisfactoryxpressionfuncertainty.The last sentence s not strictlyrue. ome writers ave considered he xioms arefullyndproduced bjections. fine ritiques Walley 1991), whowent n to constructsystem hat sesa pairofnumbers,alledupper nd lowerprobabilities,nplace ofthesingleprobability.heresult s a more omplicated ystem.My position s that hecomplicationeemsunnecessary.haveyet o meet situationn which heprobabilitypproach ppears o be inadequatendwherethe nadequacy an be fixed y employing pper nd ower alues.The pair s supposed o dealwith heprecision fprobabilityssertions; etprobabilitylone contains measure f its ownprecision. believe nsimplicity;rovidedhat tworks,hesimplers tobe preferredver hecomplicated,ssentiallyccam'srazor.With the conclusion hatuncertaintys only satisfactorilyescribed y probability,t isconveniento stateformallyhethree ules, r axioms, f theprobabilityalculus.Probabilitydepends ntwo lements: heuncertainvent ndthe onditions nderwhich ou reconsideringit. ntheextractionf balls from n urn,your robabilityor ed depends n thecondition hattheball is drawn t random.We write (A B) foryour robabilityfAwhenyou know, r areassuming, to betrue,ndwespeak fyour robabilityfA, givenB. The rules reas follows.(a) Rule1 convexity):or llA andB,0 - p(A B) - 1 andp(A A) = 1.(b) Rule2 (addition):fA andB areexclusive, ivenC,

p(A UBIC) = p(A C) + p(BIC).(c) Rule3 (multiplication):or llA,B andC,

p(ABIC) = p(A BC) p(BIC).(HereABmeans he vent hat ccursf, ndonly f,bothA andB occur.)Theconvexityule ssometimestrengthenedo includep(AIB)= 1 only f A is a logical consequence f B. Theadditions calledCromwell'sule.There s onepoint bout he ddition ule hat ppears o bemerely mathematicalicety utinfacthas importantractical onsequencesobe exhibitednSection . With he hree ules s


8/46

Philosophyf tatistics 299statedbove, t s easyto extend he ddition ulefor wo, o anyfinite umber f events. one ofthemany pproaches lready iscussed ead to therule'sholding or n infinityf events. t isusual o suppose hattdoes so holdbecause ftheundesirableesults hat ollowwithoutt.Thiscan be made xplicit ither y imply estatinghe ddition ule, rby dding fourthulewhich,together ith hethree bove, eads to addition or n infinityf exclusive vents.My personalpreferences for he atter ndto add thefollowing.

(d) Rule4 (conglomerability):f B,,} is a partition,ossiblynfinite,f C andp(AIB,C) = k,the amevaluefor ll n, hen (A C) = k.(It is easy to verifyhat ule4 follows rom ules1-3 when hepartitions finite. he definitionis due to de Finetti.) onglomerabilitys in the pirit f a class of rulesknown s 'surethings'.Roughly,f whatever appens whatever ,,)your elief s k, thenyour elief s k uncondition-ally. The assumption escribed arlier n this ection s in thesame spirit. ome statisticiansappear obe conglomerablenlywhen tsuits hem: ence hepracticalonnectiono be studiedinSection . Notethat herulesof probabilityre herenot tated s axioms n themanner oundin texts on probability. hey are deductions,part fromrule 4, fromother,more basic,assumptions.6. Significance and confidenceThereactionfmany tatisticianso the ssertionhat hey hould se probability ill be to saythat hey o italready,ndthat hedevelopmentseredescribed o nothing ore han ive littlecachet o what s already eingdone.The ournals refull f probabilities:ormal nd binomialdistributionsbound; he xponentialamilyseverywhere.tmight venbe claimed hat o othermeasure f uncertaintys used: few,fany, tatisticiansmbrace uzzy ogic.Yetthis s not rue;statisticianso use measures f uncertaintyhatdo not combine ccordingo therulesoftheprobabilityalculus.Consider hypothesis , that medical reatments ineffectual,r that specificocial factordoes not nfluencerimeevels.Thephysician,rsociologist,s uncertainboutH, and data arecollectedn thehopeofremoving,rat leastreducing,heuncertainty.statisticianalled n toadviseon theuncertaintyspectmay ecommendhat he lient ses, s a measure funcertainty,a tailarea, ignificanceevel,withH as thenullhypothesis.hat s, assuminghatH is true,heprobabilityftheobserved,r more xtreme,ata s calculated. his s a measure fthe redencethat anbe put n H; the mallerheprobability,he mallersthe redence.Thisusage flies n the faceoftheargumentsbove which ssert hatuncertaintyboutHneedsto be measured y a probabilityorH. A significanceevel s not uch probability.hedistinctionanbeexpressedtarkly:

significanceevel-theprobabilityf ome spect fthedata, ivenH istrue;probability your robabilityfH, given hedata.The prosecutor's allacy s well known n legal circles. t consistsn confusing (A B) withp(BIA), twovalues which re onlyrarelyhesame.The distinctionetween ignificanceevelsandprobabilitys almost heprosecutor'sallacy:almost'because lthough , in theprosecutorform,maybe equatedwithH, the data are treated ifferently.robabilityses A as data.Adherentsfsignificanceevels oonrecognizedhat hey ould notuse ustthedatabuthadtoincludemore xtreme' ata n theform f the ail ofa distribution.s Jeffreys1961) put t:thelevel ncludes atawhichmight avehappened utdidnot.Fromhypothesisesting,et us passtopoint stimation. parameter might e the ffectf


9/46

300 D. V.Lindleythemedical reatment,r the nfluencef the ocial factor n crime.Again 0 is uncertain,atamight e collected nda statisticianonsultedhopefully ot n that rder). he statistician illtypically ecommend confidencenterval. he developmentbove based on measured ncer-tainty illuse a probabilityensity or , andperhaps n ntervalfthat ensity. gainwe havecontrastimilaro the rosecutor'sallacy:

confidenceprobabilityhat he ntervalncludes ;probabilityprobabilityhat is includedn the nterval.The formersa probabilitytatementbout he nterval,iven ; the atter bout , given hedata.Practitionersrequentlyonfuse he wo.More mportanthan he onfusions thefact hat eithersignificanceevelsnor tatementsf confidenceombineccordingo therules ftheprobabilitycalculus.Does theconfusionmatter? t a theoreticalevel, t certainly oes, because the use of anymeasure hat oes notcombine ccording o therules f theprobabilityalculuswill ultimatelyviolate some of the basic assumptions hat were intendedo be self-evidentnd to causeembarrassmentfviolated.At a practical evel, tis not so clear and it is necessaryo spendwhile xplaininghepracticalmplications.tatisticiansend o study roblemsn solation, iththe result hat ombinationsf statementsre not needed, nd it is in the combinationshatdifficultiesanarise, s was seen n the olour-sex xamplenSection . Forexample,t s rarelypossible omake Dutch ook gainsttatementsf ignificanceevels. ome common stimatorsare known o be inadmissible. he clearest xample f an importantiolation ccurswith herelationship etween significanceevel and the samplesize n on which t is based. Theinterpretationf significantt 5%' depends n n, whereas probabilityf 5% alwaysmeans hesame.Statisticiansavepaid nadequate ttentionothe elationshipsetween tatementshatheymake andthesample izes onwhich hey re based. There re theoreticaleasonsBerger ndDelampady, 987) forthinkinghat t is too easy to obtain 5% significance.f so, manyexperimentsaisefalsehopes f beneficialffecthat oes not rulyxist.Individual tatisticaltatements,ade nisolation,maynotbe objectionable;he roubleiesin their ombinations. or example, onfidencentervals ora single parameterre usuallyacceptablebut,withmanyparameters,hey re not.Even theubiquitous amplemeanfornormal istributions unsoundn highdimensions.n an experimentith everal reatments,individualests refine utmultiple omparisonsresent roblems.cientificruths establishedby combiningheresults fmany xperiments:etmeta-analysiss a difficultreafor tatistics.How do you combine everaldata sets concerninghe same hypothesis,ach with ts ownsignificanceevel?The conclusions rom wo Student -testsnmeansY, andM2 do notcoherewith hebivariate -testoryi, a2) (Healy,1969). ncontrast,heview dopted ere asily akesthemarginsf he atteroprovideheformer.The conclusionhat robabilitystheonlymeasure funcertaintysthereforeotusta patonthebackbutstrikes tmany fthebasic statisticalctivities.avage developed is ideasin anattempto ustifytatisticalractice. e surprisedimselfydestroyingome spects f t.Let usthereforeassfrom isagreementsothe onstructivedeasthat low romhe ppreciationf thebasicroleofprobabilitynstatistics.7. InferenceThe formulationhat asserved tatistics ellthroughouthis enturys based on thedatahaving,for ach value of a parameter, probabilityistribution.his accordswith he dea that heuncertaintynthe dataneedsto be described robabilistically.t is desired o learn omething


10/46

Philosophy fStatistics 301about heparameterrom hedata. Generally otevery spect f theparameters of interest,owritetas (0, a) wherewe wish o earn bout0 with as a nuisance, o use the echnical erm.Denoting hedatabyx, theformulationntroduces(x 0, a), theprobabilityfx given and a. Asimple xamplewouldhave a normal istributionf mean0 and variance , but heformulationembracesmany omplicatedases.This handles he uncertaintyn the data to everyone's atisfaction.he parameters alsouncertain.ndeed,t s that ncertaintyhat s the tatistician'sain oncern. herecipe ays hatitalso should e described y a probability(O, a). Inso doing,we depart rom he onventionalattitude.t s often aid that heparametersre assumed o be randomuantities.his s not o. Itis the axioms that re assumed, romwhich he randomness ropertys deduced.Withbothprobabilitiesvailable, heprobabilityalculus an be invoked o evaluate herevised ncertaintyinthe ight f hedata:

p(O, x)ox (x0,a) p(O, ), (1)the onstantfproportionalityependentnly n x,not heparameters.incea is not f nterest,itcanbe eliminated,gainbytheprobabilityalculus, o give

p(Ox)= p(O,ax) da. (2)Equation 1) is theproduct ule;equation 2) the ddition ule.Togetherhey olve theproblemof inference,r, better, heyprovide frameworkor ts solution. quation 1) is Bayes'stheoremnd,bya historical ccident,thas given ts nameto the wholeapproach,which stermed ayesian.Thisperhaps nfortunateerminologys accompanied y some otherwhichis even worse. p(O) is oftencalled the priordistribution,(Olx) the posterior. hese areunfortunateecausepriorndposteriorrerelative erms, eferringothedata.Today's osterioris tomorrow's rior.The terms re so engrained hat theircomplete voidance s almostimpossible.Let us summarizehe osition eached.

(a) Statisticss the tudyfuncertainty.(b) Uncertaintyhould emeasuredyprobability.(c) Data uncertaintys someasured,onditionaln the arameters.(d) Parameterncertaintyssimilarly easuredyprobability.(e) Inferencesperformedithinhe robabilityalculus,mainly y quations1) and 2).Pointsa) and b) havebeen discussed nd c) is generallyccepted. ointe) follows roma)-(d). The mainprotestgainstheBayesian osition as been opoint d). It s thereforeonsiderednext.8. SubjectivityAtthebasiclevelfromwhichwe started,t s clear hat neperson's ncertaintiesreordinarilydifferentrom nother's. heremaybe agreementverwell-conductedamesofchance, o thattheymaybe usedas a standard,utonmany ssues, ven nscience, here an be disagreement.As this sbeingwritten,cientistsisagree nsome ssuesconcerningenetically odified ood.Itmighthereforee sensible o reflecthe ubjectivitynthenotation. hepreferred ayto dothis s to include he oncept f a person's nowledget the ime hat heprobabilityudgmentsmade.Denotinghat y K, a better otation hanp(O) is p(OIK), theprobabilityf 0 givenK,changingop(Olx,K) onacquiringhedata.This notations valuablebecause temphasizeshe


11/46

302 D. V. Lindleyfact hatprobabilitys alwaysconditionalsee Section5). It depends n two arguments:heelementwhose uncertaintys being described nd the knowledge n which hat ncertaintysbased. The omission f the conditioningrgument ften eads to confusion. he distinctionbetween rior nd posteriors better escribed y emphasizinghe differentonditions nderwhich heprobabilityf heparameters being ssessed.It has been suggested hattwopeoplewiththe same knowledge hould have the sameuncertainties,nd thereforehe ameprobabilities.t is called thenecessary iew.There re twodifficultiesith his ttitude. irst, t is difficulto say what s meant y twopeople having hesameknowledge,nd also hard o realize npractice. he secondpoint s that,ftheprobabilitydoes necessarily ollow,tshould e possible o evaluate t without eferenceo a person. o farno-one as managed he valuationnan entirelyatisfactoryay.One way s throughhe onceptof gnorance.fa state f knowledge was identified,hat escribed he ack of knowledgebout0, and that (O I) was defined,hen (OIK), for nyK, could be calculated y Bayes's heorem(1) on updating to K. Unfortunatelyttemptso do this ordinarilyead to a conflict ithconglomerability.or example, uppose hat is known nly o assumepositive nteger aluesandyou re otherwisegnorantbout . Then heusualconcept f gnorancemeans ll values reequally robable: (O = i I) = c for ll i. The addition uleforn exclusive vents 0 = i}, withne> 1, meansc = 0 since no probabilityan exceed1by convexity. ow partitionhepositiveintegersnto etsof three, ach containingwo odd, nd one even,value:A,, (4n - 3, 4n - 1,2n) will do. If E is the event hat0 is even,p(EIA,,)= 3 and by conglomerability(E) = 3Another artition,ach set with wo even and one odd value, sayB,,= (4n - 2, 4n, 2n - 1),has p(EIBn) 2 and hence p(E) =2 in contradictionith he previous esult. y the suitableselection fa partition,(E) can assume nyvalue n theunit nterval.uch a distributions saidtobe improper.

Unfortunatelyost ttemptsoproduce (OII) bya necessary rgumentead to improprietywhich,naddition oviolating onglomerability,eads to otherypes funsatisfactoryehaviour.See, for xample, awidetal. (1973). The necessary iewwasfirstxaminedndetail yJeffreys(1961). Bemardo 1999) and others avemaderealprogressut he ssue s still nresolved. eretheviewwill be taken hat robabilitys an expression y a personwith pecified nowledgeabout n uncertainuantity.hepersonwillbereferredo as 'you'. p(A B) isyour elief boutAwhenyouknowB. In thisview, t s incorrectorefer o theprobability;nly oyours.t is asimportantostate he onditionss it stheuncertainvent.Returningo theexample f0 taking ositive nteger alues, s soonas youunderstandhemeaning ftheobject o which heGreek etterefers, ou re,becauseof that nderstanding,olonger gnorant. hen p(O = i) = ai with ai = 1. Some statistical esearchs vitiated y aconfusion etween heGreek etterndthe ealityhattrepresents.challenge nyoneoproducea realquantityboutwhich hey retruly gnorant. furtheronsiderations that computercouldnotproduce neexample f 0. Sincep(O - n I) = nc= 0, p(O> n) = 1,so 0 must urelybe largerhan nyvalue hat ou are oname, rthe omputeran handle.A further atterequiresttention.t is common o use the ameconcept nd notation henpart fthe onditioningvents unknownut ssumedo betrue. or xample,ll statisticianssep(xI0,a) as above,or,more ccurately,(xI0,a, K). Heretheydo notknow hevalueof theparameter. hat sbeing xpressedsuncertaintybout , ftheparametersere o havevalues0anda (and they adknowledge ). It s notnecessaryodistinguishetweenuppositionnd factinthis ontextndthenotation (A B) is adequate.Thephilosophical ositions thatyourpersonal ncertaintys expressed hrough our rob-ability f anuncertainuantity,ivenyour tate fknowledge,eal or assumed.This s termedthe ubjective,rpersonal,ttitudeoprobability.


12/46

Philosophy fStatistics 303Many people, especially n scientificmatters, hink hat their statementsre objective,expressed hroughheprobability,ndare alarmed ythe ntrusionf subjectivity.heir larmcanbealleviatedyconsideringealityndhow hat ealitysreflectedn theprobabilityalculus.

We discuss nthe context f sciencebut he approach ppliesgenerally.aw providesnotherexample. upposethat0 is thescientificuantityf interest.In criminalaw,0 1 or 0 0accordingo whetherhedefendantid, r didnot, ommithecrime.)nitiallyhe cientist illknowittle bout0, becausetherelevantnowledgeaseK is small, nd two cientists ill havedifferentpinions,xpressedhroughrobabilities(0IK). Experimentsillbe conducted,ataxobtained nd their robabilities pdated o p(0lx,K) in the wayalreadydescribed.t can bedemonstratedsee Edwards t al. (1963)) that nder ircumstanceshat ypicallybtain, s theamountf data ncreases,hedisparateiewswillconverge,ypicallyo where is known,r atleast determined ith considerable recision. his is what s observed n practice:whereasinitiallycientistsaryntheir iews nd discuss, ometimesigorously,mong hemselves,heyeventuallyometo agreement. s someone aid, heapparentbjectivitys really consensus.There s thereforeood agreementerebetweencientificracticend the Bayesian aradigm.There re caseswhere lmost ll agree n a probability.hesewillbediscussedwhenweconsiderexchangeabilitynSection 4.It is now possible o see why here as been a reluctanceo acceptpoint d), the use of aprobabilityistributiono describe arameterncertainty.t is becausetheessential ubjectivityhas notbeen recognized.With ittle ata,p(O,a) varies mong ubjects: s the data ncrease,consensussreached. otice hat (x 0, a) is alsosubjective.his sopenly ecognized hen wostatisticiansmploy ifferentodelsn theirnalysis fthe amedata et.We shallreturnothesepointswhen he oleofmodels streatedn Sections and 11.9. ModelsThetopicofmodelshas been carefullyiscussed y Draper 1995) from heBayesian iewpointand thatpapershouldbe consulted or more detailed ccount han hatprovided ere. Thephilosophicalosition eveloped ere s that ncertaintyhould e described olely n terms fyourprobability.he implementationf this idea requires he constructionf probabilitydistributionsor ll theuncertainlementsn thereality eing tudied. hecomplete robabilityspecification ill be called a (probability) odel, though heterminologyiffers rom hatordinarilysed nstatistics,n a wayto be describedater. t also differsrommodel s used nscience.The statistician'sask s toconstructmodelfor heuncertain orld nder tudy. avingdonethis, heprobabilityalculus nables he pecific spects f nterestohavetheir ncertain-ties computedn theknowledgehat s available.There rethereforewoaspects o ourstudy:theconstructionf themodel nd the nalysis f thatmodel.The latters essentiallyutomatic;in principlet can be doneon a machine. he formerequires lose contactwithreality. oparaphrasendexaggerateeFinetti,hink hen onstructinghemodel;witht,donot hinkutleave tto the omputer. erepeat hepoint lreadymadethat,ndoing his, he ubject,whoseprobabilitiesrebeing ought,he you' inthe anguage dopted ere, s not he tatistician,uttheclient, ften scientist ho has asked for statisticaldvice.The statistician'saskis toarticulatehe cientist'sncertaintiesnthe anguage fprobability,nd then ocompute ith henumbers ound.A model s merely ourreflectionfreality nd, ikeprobability,t describesneither ounortheworld, utonly relationshipetween ouandthatworld. t is unsound orefer o the ruemodel.Onetime hat hisusagecanbe excused swhenmostpeopleareagreedon themodel.Thus themodelof theheights f fathers nd sons as bivariate ormalmightreasonablyedescribeds true.


13/46

304 D. V. LindleyWhatuncertaintiesre therena typical cenario? hefundamentalroblem f nferencendinductions to use past data to predict uture ata. Extensive bservationsn themotions fheavenly odiesenables heir uture ositions o be calculated. linical tudies n a drug llowa

doctor o give prognosis or patient orwhom hedrug s prescribed.ometimesheuncertaindata re nthepast,not hefuture. historian ill use what vidence e has to assesswhatmighthavehappenedwhere ecords re missing. court f criminalaw enquires boutwhathad hap-penedon thebasis of ater vidence.We shall,however,se the emporalmage,with astdataxbeingusedto nfer uture ata y (as x comesbefore in the lphabet).n this iew, he ask s toassess p(ylx,K). In the interests f clarity,he background nowledge, ixed hroughouthetreatment,illbe omitted rom henotationndwe write (ylx).One possibilitys to try o assessp(ylx) directly.his s usually ifficult,houghtmaybethoughtf as thebasis of the pprenticeshipystem. ere an apprentice ould it t themaster'sfeet nd absorb hedatax. Withyears f such xperience,he pprenticeould nfer hatwouldbe likely o happenwhenhe worked n his own. Successive bservationn theuse ofash n theconstructionf a wheelwould nablehim o employ sh forhis ownwheel.There s, however,better aytoproceed nd that s to studyhe onnectionsetween andy, andthemechanismsthat perate.Newton's aws enable the tidesto be calculated.Materials cience assists n thedesign nd constructionf a wheel.Mostmodernnferencean be expressed hroughparameterO that eflectsheconnection etweenhetwo sets of data. Extendingheconversation,trictlywithin he robabilityalculus, o nclude ,p(YIx) = 0IH,x)p(OIx) dO.

It s usual to suppose hat, nce0 is known, hepast data are rrelevant.nprobabilityanguage,given , x and y are ndependent.ombininghis actwith hedeterminationfp(Olx)byBayes'stheorem, e have

P(ylx) p(yI0) p(xI ) p(O)dO fp(x 0) p(O)dO.Now the ask s toassesstheuncertainty(O) about heparameterndthe wodatauncertainties,p(xI0) andp(yI0),given heparameter.ften he nferenceanstop tp(Olx), eaving thersoinsert (yI0). Thismight appenwith hedrug xample bove,where hedoctorwouldneed toknow hedistributionfefficacy and hen ssess p(y 0) for he ndividual atient.Althoughmuch nferences rightlyxpressednterms fthe valuationfp(Olx), here s animportantdvantagen contemplating(ylx). The advantage ccruesfromhefact haty willeventuallye observed; hedoctorwill see whathappens o thepatient. he parameters notusually bserved. heuncertaintyf0 often emains;hat fy disappears. his feature nablesthe ffectivenessfthe nferenceo be displayed yusing scoringule nanextendedersion fthat escribednSection . Ifthe nferences p(ylx)andy is subsequentlybserved obe Yo,score function{yo,p( Ix)} describes owgood the nference as, so that he client nd thestatisticianave their ompetencesssessed.The method asbeen usedinmeteorology,.g. inforecastingomorrow'sainfall. uch methodsrenotreadily vailableforp(Olx).One ofthecriticismshathasbeen evelled gainst ignificanceevels s that ittle tudy as been made ofhowmanyhypotheses,ejected t 5%, have subsequentlyeen shown o be true.There s noreason o think hat t s 5%.Theory uggestshat tshould emuchhigherndthat ignificancestoo easilyattained.A weather orecaster ho predicted ainon only5% of days,whenitsubsequentlyained n20%,wouldnotbehighlysteemed. he bestway o assess the uality f


14/46

Philosophyf tatistics 305inferencess to checkp(Olx)throughhedataprobabilities(ylx)that hey enerate.As previously entioned,t s usually ecessaryo ntroduceuisance arameters, in additionto 0, to describe dequately heconnection etween and y, and to establish he ndependencebetween hem, iven 0, a). In the drug xample, might nvolvefeatures f the ndividualpatient. uisance arametersmpose ormidableroblems or omeforms f nference,ike hosebased solely n likelihood, ut, n principle,re easilyhandledwithin heprobabilityrameworkbypassing rom heointdistributionf 0, a) tothemarginal or : equation2).The ntroductionfparameterseduces he onstructionf a model o providing(x 0, a) andp(O, a). p(yI0,a) also arisesbut ts ssessments similar o that or and neednotbe separatelydiscussed.Herewe see thedistinctionetweenur use of model' andthat ommonlydopted,where nly he data distribution,iven heparameters,s included.Our definitionncludes hedistributionftheparameters,ince hey orm n mportantart ftheuncertaintyhat s present.Mostofthe urrentiteraturen models hereforeoncerns hedata and s discussednthenextsection. or hemoment, e ust repeat hepointmadeearlier hat venp(xI0,a) is subjective.common eason orwronglyhinkinghatt s objective ies in thefact hat here s oftenmorepublic nformationn thedata thanon the parameters,ndwe saw in Section8 that,withincreasednformation,eople end o approach greement.Whygo throughheritual fdetermining(xI0,a) andp(O, a), and then alculating(Olx)?If p(O, a) can be assessed,whynot ssessp(Olx)directlynd avoid omecomplications?ouseterminologyhat do not ike: fyour rior an be assesseddirectly, hynotyour osterior?artof he nsweries n the nformationhat s typicallyvailable bout hedatadensity,ut hedesirefor oherences themajorreason.A set ofuncertaintytatementss said tobe coherentftheysatisfyhe rules of theprobabilityalculus. Thus,thepair of statements(AIB) = 0.7 andp(Al -B) = 0.4 do notcoherewiththepairp(BIA) = 0.5 and p(BI -A) = 0.3. (Here ,Bdenotes he omplementfB.) Think fA as a statementboutdatax andB as a statementboutparameter. The firstairrefersouncertaintiesnthedata andcohereswith hefirstarameterstatement,(BIA) = 0.5, fordataA. (Take p(B) = 0.4/1.1= 0.36.) But all three o notcoherewith he econdparametertatementordata -A, that (BI r-A)= 0.3. Withp(B) = 0.36,thecoherentalue s 0.22.Thestandardrocedurensureshat ou repreparedor nyvaluesofthedata,A or -A, and hefinal nferencesboutB willcollectively ake ense.10. Data analysisMuch statistical ork s not concernedwith a mathematicalystem,whether requentistrBayesian,butoperates t a less sophisticatedevel.Whenfaced witha new set of data,astatisticianill play round'with hem,nactivityalled exploratory)ata nalysis. lementarycalculationswill be made; simple graphswill be plotted. everal valuable deas have beendeveloped or playing', uchas histogramsndbox plots.We arguethat his s an essential,importantnd worthwhilectivityhat its ensiblynto hephilosophy.he view dopted ere sthat ataanalysis ssistsntheformulationfa model nd s anactivityhat recedesheformalprobabilityalculationshat reneeded ornference.heargumentevelopedo farnthis aperhas demonstratedhe need forprobability.ata analysisputsfleshontothismathematicalskeleton. heonlynovelties hatwe add to conventionalataanalysiss therecognitionhat tsfinal onclusionshould e intermsfprobabilityndshould mbrace arameterss wellas data.In the anguage f he ast ection,he onclusionsfdata nalysishould ohere.

The fundamentaloncept ehind hemeasurementfuncertaintyas thecomparison ithstandard.uchcomparisonsre often ifficultndtheres a needto find omereplacement. edo notmeasure ength y usingkryptonight, hestandard,utemploy thermethods. ata


15/46

306 D. V.Lindleyanalysis nd theconcept f coherences sucha replacement.uppose hatyou need to assess asingle robability;hen ll youhave o guideyou s thenecessityhat hevalue ies between and1. ncontrast,uppose hat heneed s to assess several robabilitiesfrelated vents rquantities,when hewholeof therich alculus f probabilitiess available o helpyou n your ssessments.In the xample hat oncluded ection , youmight avereached hefour aluesgiven here, utconsiderationsf coherencewould forceyou to alter t least one of them.Coherence cts ikegeometrynthemeasurementf distance;tforces everalmeasurementso obey he ystem.Wehave eenhow hishappens n replacing (y x) by p(x 0, a) and p(O, a). Let us consider his nditsrelationshipith ata nalysis, onsideringirsthedatadensity (x 0, a).A familiarnd useful ool here s thehistogramnd modem ariantsike stem-and-leaflots.These helpto determine hether normal ensitymight e appropriate,r whetheromericherfamilys required.fthedata onsist f wo, rmore, uantities = (w, z), then plot f againstw will helpto assess theregressionf z on w and hencep(zlw,0, a). These devices nvolve heconcept frepeated bservations,.g. to constructhehistogram. e shallreturno this ointndiscussion fthe oncept fexchangeabilityn Section 4.There re issues herethathave not alwaysbeen recognized. ou are making n uncertaintystatement,(x 0, a), for quantity, which,with hedataavailable,s foryou certain.Moreoveryouaredoing twith ealthought,n theform f dataanalysis boutx. It is strange nly o useuncertaintyprobability)ortheonlycertain uantity resent. urthermore,upposethatt(x)describes he spects f thedatathat ouhave considered,hehistogramr theregression. hentheresult fthedata nalysis s really {xl0, a, t(x)}; you re conditioningn t(x). For xample,you might ay thatx is normalwithmean0 and variance , but only after eeing t(x), orequivalently oingthe data analysis.Thismaylead to spurious recisionn the subsequentcalculations. newaytoproceedwouldbe to constructhemodelwithoutooking t thedata.Indeed, his snecessary hen esigninghe xperimentSection16). The constructionouldonlycomein close consultation ith heclient nd would nvolve argermodels han re currentlyused.Perhaps ataanalysis anbe regardeds approximatenference,learing utthegrosseraspects fthe argermodel hat renotneedednthe perational,mallermodel.Anotheroint s that (xJH, ), say ntheform f a histogram,s only xhibited or nevalueof 0, a), namelyheuncertainalue hat olds here. hedatacontainittle vidence hat,ven fx - N(0O, ao), it s N(0, a) insituationsnobserved.heres a case thereforeormakingmodelsas big as yourcomputing owerwill accommodate,o allow fornon-normalitynd generalparameteralues.The sizeofa model s discussednSection 1. Notice hat hedifficultiesaisedinthe ast woparagraphsre as relevanto thefrequentists theyreto theBayesian.Theassessmentroblems differenthen tcomestotheparameterensityecause theresoften orepetitionndthefamiliaroolsofdata nalysisrenolongervailable. urthermore,nhandlinghedatadensity,everal tandard odels rereadily vailable, .g.the xponentialamilyand methods uilt roundGLIM. Thesemodelshaveprimarilyeendesigned or ase ofanalysisthroughhepossession fspecialpropertiesike sufficienttatisticsf fixedowdimensionality,though hey avethedifficultyhat utliers renot asily ccommodated.heseconstraintsavebeen imposedpartly hroughimitationsf computer apacity ut more mportantlyecause,within hefrequencypproach,here renogeneral rinciplesnd a new modelmayrequireheintroductionf new ideas. Moderncomputationalechniquesessen the first ifficultyndBayesianmethods,withtheirubiquitous se of theprobabilityalculus,remove he secondentirely;he bject s always o calculate (Olx).We shall eturno this ointnSection 5.

Fewstandard odels re available or heparameterensity,ssentiallyimitedo thedensitiesthat re conjugate o the member f theexponential amily hosen for he datadensity. hefrequentisthant s 'where did you get thatprior?'. t is not a silly gibe; there re serious


16/46

Philosophyf tatistics 307difficultiesut hey repartly ausedby failure o ink heoryndpractice. haveofteneenthestupid uestionposed 'what s an appropriateriorfor the variance 2 of a normal data)density?'.t s stupid ecause is usta Greek etter. o find heparameterensity,t s essentialto go beyond he alphabet nd to investigatehereality ehind r2. What s it thevariance f?What ange fvaluesdoes the lient hinkthas? Recall that he tatistician'sask s toexpressheuncertaintyf you, he lient,nprobabilityerms. sensible orm fquestionmight e 'what syouropinion boutthe variabilityf systolic lood pressuren healthy, iddle-agedmalesinEngland?'.But, venwith areful egard or ractice,t wouldbe stupid o deny he xistencefvery eal, nd argely nexplored,roblems ere.This s especially ruewhen, s inmost ases,theparameterpace has highdimensionality.e are lacking n methods f appreciating ulti-variate ensities.This s true f dataas well as parameters.)hysicists id not denyNewton'slaws because several f the deas that e introduced eredifficulto measure.No, they aid thatthe aws made sense,theyworkwherewe can measure, o let us developbettermethods fmeasurement.imilar onsiderationspply o probability. neglected rea of statisticalesearchis theexpression f multivariatepinion n terms f probability,here ndependences invokedtoo often, n grounds f simplicity,gnoring eality.t is notoften ecognized hat henotion findependence,ince t nvolves robability,s also conditional. he mantra hatxl,x2, ,xJforming random ample are ndependents ridiculous hen hey re used to infer ,,1.Theyare ndependent,iven .It is sometimes rgued hatdataanalysis an make no contributiono theassessment f adistributionor heparameterecause t nvolves ooking tthedata,whereaswhat s needed s adistributionrior o thedata.This s countered y theobservationhatwe all use datatosuggestsomethingndthen onsider hat ur ttitudeo twas withouthedata.You see a sequence fOsand Is and notice ew, ut ong, uns.Couldthe equence e Markov nstead fexchangeablesyouhad anticipated?ou think bout reasonsfor hedependence nd,havingdecidedthatMarkov hain spossible, hinkbout tsvalue. Hadyou seen Is onlywhen heorderwasprime,youwould ail o find easons ndaccept he xtraordinaryhinghat ashappened.11. Models againA model s a probabilisticescriptionf a client's ituation,hose ssessments helpedbydataanalysisndexplorationf the lient's resent nderstanding.everal roblems emain,f whichone s the ize ofthemodel. houldyou nclude xtra uantities,esidesx,as covariates?houldtheparametersncreasennumber oofferreater lexibility,eplacing normal istributionyaStudent's, say? Savageoncegavethewiseadvicethat model hould e as bigas anelephant.Indeed, he deal Bayesianhas one modelembracingverything:hathas been termed worldview.Such a model s impracticalndyoumust e content ith smallworld mbracing ourimmediatenterests. ut howsmallshould t be? Reallysmall worldshave theadvantage fsimplicitynd thepossibilityfobtaining any esults,ut hey avethedisadvantagehat heymaynotcapture ourunderstandingfrealityo thatp(ylx) based on themmayhave a highpenaltycore.Compromises calledfor, ut lways hoose he argestmodel hat our omputa-tionalpowerswill tolerate. ne successfultrategys to use a largemodelandto determine,throughobustnesstudies, hat spects fthemodel eriouslyffectour inal onclusion. hosethat o not an beignorednd somereductionnsize achieved.It is valuable o thinkbout herelationshipsetween he mallworld elected nd the argerworlds hat ontain t. In England t is currentractice o publish eague tables of schools'performanceshat se only xaminationesults.Manycontend hat his s a ridiculouslymallworld ndthat theruantities,ike heperformancefpupils tadmission,hould e included.t


17/46


18/46

Philosophyf tatistics 30912. OptimalityThe position as beenreached hat hepractical ncertaintieshould edescribed yprobabilities,incorporatedntoyourmodel and thenmanipulatedccording o the rulesof the probabilitycalculus.We now consider he mplicationshat hemanipulations ithin hat alculushave onstatisticalmethods, specially n contrast ith frequentistrocedures,hereby xtending hediscussion f ignificanceests ndconfidencentervalsnSection . It s sometimesaid, y hosewho use Bayes estimatesrtests, hat ll theBayesian pproach oes is to add a prior o thefrequentistaradigm. prior s introduced erely s a devicefor onstructingprocedure,hat sthennvestigated ithin hefrequentistramework,gnoringhe adder f theprior ywhich heprocedure as discovered. his s untrue:he adoption f the fullBayesianparadigm ntailsdrastichangentheway hat ou hinkbout tatistical ethods.A large mount f effort as beenput nto hederivationf optimumests nd estimates. hisis evident n thetheoreticalide where hesplendid cholarly ooks ofLehmann 1983, 1986)are largely evoted o methods f finding ood estimates nd testsrespectively.gain,moreinformally,ndataanalysis, easons readvanced or sing ne procedureatherhan nother,swhen rimmed eans re rightlyaid to be better han awmeans nthepresence f outliers. etus thereforeook at nference,nthe enseofsaying omethingbout parameter, given atax,in thepresence f nuisanceparameters. The frequentist ay seek the bestpointestimate,confidencentervalrsignificanceest or .A remarkable,nd largely nrecognized,act s that,withinheBayesianparadigm, ll theoptimalityroblemsanish; whole ndustryisappears. ow canthis e? Consider herecipe.tis to calculate (Olx,K), thedensity or heparameterf nterestiven hedata and backgroundknowledge. his densitys a complete escriptionf your urrent nderstandingf0. There snothing ore o be said. t s anestimate: our nly stimate.ntegratedver setH, itprovidesyour ntire nderstandingf whether is true.There s nothing etter hanp(Olx,K). It isunique; heonly andidate. onsider he ase ofthe rimmed eansustmentioned.fthemodelincorporatesimplenormality,he density or0 is approximatelyormal boutx-, hesamplemean.However,uppose hat ormalitysreplaced yStudent's with wonuisance arameters,spread nddegrees ffreedom);hen hedensityor will be centred,oton-x, utonwhat sessentially trimmed ean. n otherwords, he estimate rises nevitablynd notbecause ofoptimalityonsiderations.The Bayesian's niqueestimate,heposterioristribution,epends n theprior,o there ssomesimilarityetween heBayesian nd thefrequentisthouses a prior o constructheiroptimumstimates.The class of good frequentistroceduress theBayes class.) The realdifferences that hefrequentistilluse differentriteria,ike he rrorate, atherhan oherenceto udgethe ualityf he esultingrocedure.his s discussed urthernSection 6.13. The likelihood principleWe have een hat arametricnferencesmadebycalculating

p(O x) = p(x 0) p(O) fp(x 0) p(O)dO. (3)Consider (x 0) as a functionftwoquantities,and 0. As a functionfx,for nyfixed ,p( 0)is a probabilityensity,amelyt s positivend ntegrates,verx,to 1. As a functionf0, foranyfixed , p(x .) ispositive utdoes notusually ntegrateo 1. t s calledthe ikelihood f0 forthe fixed . It is immediaterom quation3) that heonlycontributionhat he data make toinferencesthroughhe ikelihood unctionor he bserved .This s the ikelihoodrinciplehat


19/46

310 D. V.Lindleyvaluesof x, otherhan hat bserved, laynorole n nference.valuable eferencesBergerndWolpert1988).This facthas importantecognizedonsequences.Wheneverninferencen integrationakesplace overvalues of x, the principles violated nd theresultingroceduremaycease to becoherent. nbiasedestimatesnd tail area significanceests are among thecasualties.Thelikelihoodunctionhereforelays moremportantole nBayesian tatisticshantdoes nthefrequentistorm, et ikelihoodlone s notadequate ornferenceutneedsto be tempered ytheparameteristribution.ncertainty ust e describedyprobability,ot ikelihood. eforeenlargingnthis emark,t s importanto be clearwhat s meant y ikelihood.f a modelwithdatax hasbeendevelopedwith arameters0, a), then (x 0, a) as a functionf 0, a), for hefixed bserved alue of x, s undoubtedlyhe ikelihoodunction.owever,nferencenequation(3) doesnot nvolvehe ntireikelihoodunction,ut nlyts ntegral

p(xI0) =Jp(x IH,a) p(a I ) da. (4)We refer o this s the ikelihood f0 but he erminologysnot lways ccepted. hereason sclear: tsconstructionnvolves neaspect, (a 0), oftheparameterensity,(O,a), which atteris not admitted o the frequentistr likelihood chools. n neitherchool is theregeneralagreementbout whatconstituteshe ikelihood unction or parameter of interestn thepresence f a nuisance arameter. There reat leasta dozencandidatesnthe iterature.orexample,naddition othe ntegratedormnequation4), theres p(x 0, a), where is thevaluethatmakesp(xI0, a) over a maximum. heplethora f candidateseflectshe mpossibilityfany atisfactoryefinitionhatvoids he ntrusionfprobabilitiesor heparameters.The reasonfor ikelihood eing, n its own, nadequate s that, nlikeprobability,t is notadditive.fA andB aretwoexclusive ets, henp(A U B) = p(A) + p(B), omittinghe condi-tions,whereast s nottrue hat (A U B) = l(A) + I(B) for likelihood unction(.). Since thepropertiessedas axioms nthedevelopmentf inference,.g. in thework fSavage, ead toadditivity,nyviolationmay ead tosomeviolation fthe xioms.Thishappenswith ikelihood.In Section we had an example nvolvingolour nd sex,whichwas expressednterms ftheinformaloncept foneevent eingmore ikely han nother.n fact, heexampleholdswhen'likely' s used n the echnicalense s defined ere.Likelihoods anessentialngredientntheinferenceecipe ut tcannot ethe nly ne.Notice hat he ikelihoodrinciplenly pplies o nference,.e. tocalculations ncethedatahave been observed.Beforethen, .g. in some aspects of modelchoice,in the designofexperimentsr in decision nalysis enerally, considerationf several ossibledatavalues sessentialsee Section 6).14. FrequentistconceptsEversincethe1920s, tatistics as beendominatedythefrequentistpproach ndhas, byanysensible riterion,eensuccessful; etwe have seenthat t clasheswith he coherentiewinapparentlyeriousways.How canthisbe? Ourexplanations that heres a property,hared ybothviews, hat inks hemmore losely han hematerialo farpresented eremightuggest.The link s theconcept f exchangeability. sequence xl,x2, .., x") ofuncertainuantitiesis, foryou,exchangeable nder onditions ifyour ointprobabilityistribution,ivenK, isinvariant nder permutationf the suffixes. or example,p(x1= 3,x2= 51K)= p(x2= 3,xi = 51K)onpermutingand2. An nfiniteequences exchangeablefevery initeubsequenceis so udged.The roles f you'andK havebeenmentionedoemphasizehat xchangeabilitys a


20/46

Philosophy fStatistics 311subjectiveudgmentndthat oumay hange our pinionf he onditionshange.If you udge a sequence o be (infinitely)xchangeable,hen our robabilitytructureor hesequence s equivalent ointroducingparameter,/ay, uch hat, ivenV/, hemembers f thesequence re ndependentnd denticallyistributedIID). Astheparameters uncertain,ouwillhavea probabilityistributionor t. This result s due to de Finetti1974, 1975). Ordinarily/will consistof elements 0, a) of which0 is of interest nd a is nuisance.Consequentlyexchangeabilitymposes hestructuresed above but with he addition hat he data x havetheparticularorm f ID components.urthermore,p is related o frequencyropertiesf thesequence.Thus, n the simple ase where i is either or 1, the Bernoulli equence, is thelimitingroportionf them hat re 1. Consequently,Bayesianwho makes he xchangeabilityjudgments effectively aking he same udgment bout data as a frequentist,ut withtheaddition fa probabilitypecificationor he arameter.The concept f ID observationsas dominatedtatisticsn this entury.ven when bviouslyinappropriate,s in the study ftime eries, he modelling ses IID as a basis. For example,x- Ox,-,maybesupposedID for ome0, leading oa linear, utoregressive,irst-orderrocess.Withinhe ID assumption,requencydeasare pposite,ome venwithin heBayesian anon, othere as developed belief hat ncertaintyndprobabilityre thereforeasedon frequency.Some statisticsexts nly eal with ID data ndthereforeestricthe ange fstatisticalctivities.Their xampleswill come fromxperimentalcience,where epetitions basic, nd notfromaw,wheret s not.Frequency, owever,s not dequatebecause there s ordinarilyo repetitionfparameters;heyhave uniqueunknown alues. Consequentlyheconfusion etween requencyandprobabilityas denied thefrequentistheopportunityf usingprobabilityorparameteruncertainty,ith he esult hat t has beennecessaryor hem odevelopncoherentonceptsikeconfidencentervals.

The use of frequencyoncepts utside xchangeabilityeadstoanother ifficulty.requentistsoften upportheir rgumentsy saying hat hey re ustifiedin the ong run',to which hecoherentesponse s 'what ong run?'. Forexample, confidencentervalsee Section6) willcoverthetruevalue a proportion - a of times n the ongrun.To make sense of this t isnecessary o embedthe particular ase of data x intoa sequenceof similar ata sets: whichsequence?;what s similar? heclassicexamples a data etconsistingfrsuccessesnntrials,judged o be Bernoulli.nthe equence o wefixn,orfixror someother eaturefthe bserveddata? tmatters.ayesians roviden answer or he ingle ituation, hereas requentistsftenneed oembed he ituationnto sequence fsituations.The restrictionfprobabilityofrequencyan ead to misrepresentations.ere s an example,concerninghe determinationfphysical onstants,uch as thegravitationalonstant . It iscommon nd reasonable o supposethat hemeasurements ade at one place and time areexchangeablendunbiased,achhaving xpectation . It is reasonable o use theirmean s thecurrent stimate f G. Some rejection f outliersmaybe needed before his s done. Theuncertaintyttachedo this stimates found y taking 2, equal to theaverage f thesquareddeviations rom hemean, nd quoting standard rror f s/IVn,wheren is thenumber fmeasurements.his leads to confidenceimits orG. Experiencehowsthat he more recentestimatessuallyie outside heconfidenceimits f earlier stimates.n otherwords, he imitswere oonarrow. scoring ulefor stimatorsf G wouldproduce largepenalty core.Thereason sthat hemeasurementsreactually iased.Sincethe mount fthebiasis not menabletofrequencydeas, t is ignored. heBayesian pproachwouldhave a distributionor hebiasandwoulduse as a prior or G theposteriorromhe astestimate, ossibly djusted or nymodificationsnthemeasurementrocess.Often tandardrrorsretoo smallbecauseonly heexchangeable omponent f uncertaintys considered. imilarmistakes an arise withthe


21/46

312 D. V. Lindleypredictionsffuture umbers f cases ofacquired mmune eficiencyyndrome.hey an gnorechangesnpersonal ehaviour r Governmentolicy, hanges hat renot menable o frequentistanalysis.15. Decision analysisIt has been notedhow statistics eganwith he collection ndpresentationf data, and thenextendedo nclude he reatmentfthedata nd theprocesswhichwe nowcall inference.hereis a furthertagebeyond hat, amely he use of data, nd the nferencesrawn rom hem, oreach decisionndto nitiatection.nmyview, tatisticiansave realcontributionomake odecision nalysis nd should xtend heir ata collection nd inferenceo include ction.Themethods fRamsey ndSavagehavedemonstratedow hefoundationsan bepresentedhroughdecision nalysis. heextensiono include ction an be better nderstoodfwe ask what s thepurpose f an inferencehat onsistsncalculating (ylx) for uture atay,conditionalnpastdata x. Anexample itedwas a doctorwho haddata on a drug nd wished o inferwhatmighthappen o a patient iventhe drug.The example nvolves decision,namelywhichdrug oprescribe,nother rugpossibly eadingto a differentnference ory. We argue,followingRamsey, hat n inferences only f value f t s capableof beingused toinitiate ction. artialknowledgehat annot e used s of ittle alue.Even n tsparametricorm, (Olx)will onlybeworthwhilef tcan be incorporatednto ctions hat nvolve heuncertain. Marxwasright:hepoint s notusttounderstandheworldinference)ut lso to changet action). et us see howthis an be done ntheBayesian iew.Thestructuresedby Savage ndotherss toformulatelist fpossible ecisions thatmightbe taken. heuncertaintys capturedn a quantityor parameter). Thepair d, 0) is termedconsequence, escribing hatwillhappenfyoutakedecision when heparameteras value 0.Wehave eenhow he ncertaintyn0needs obe describedy probabilityistribution(O).Thiswill be conditionalnyour tate fknowledge, hich s omittedrom henotation.tmay lsodepend n thedecision,s inthe ase where hedecisions reto nvestnadvertising,rnot, nd0is next ear's ales.Wethereforerite (O d). Thefoundationalrgumentoesonto show hat hemerits f he onsequenced, 0) canbedescribedy realnumber(d,0),termedhe tilityf heconsequence. neconsequencespreferredo anotherf thasthehighertility.f hese tilitiesreconstructedna sensible ay,he estdecision s thatwhichmaximizes ourxpected tility

fu(d, 0) p(O d) dO.Theaddition fa utility unctionor onsequences,ombinedwith heprobabilityescriptionfuncertainty,eadstoa solution o thedecision roblem. tility as to be described ith are. t snotmerely measure f worth, uta measure f worth n a probabilitycale. If the bestconsequence asutility and worst tility , then onsequence d, 0) has utility (d, 0) ifyou(notice he ubjectivelement)re ndifferentetween

(a) the onsequenceor ure nd(b) a chanceu(d, 0) ofthebest and1 u(d, 0) oftheworst).It is thisprobabilityonstructionhat nablestheexpectationo emerge s theonlyrelevantcriterionor he hoiceofdecision.Utilitymbraces ll aspects fthe onsequence. orexample,ifoneoutcome f a gamble s a winof?100, tsutilityncludes otonly n increasenmonetaryassets but also the thrill f thegamble.Some analyses, ased solelyon money,re defectivebecauseoftheirimited iewofutility.


22/46

Philosophy fStatistics 313Notice hat,ust as p(O) is not he tatistician'sncertainty,utratherhe lient's, o theutilityis that f thedecisionmaker. he statistician'sole s to articulateheclient's referencesn theform f a utility unction,ust as it s to express heir ncertaintyhroughrobability.otice lso

that he nalysis upposes hat heres only ne decisionmaker, he you' ofour ext, houghyou'maybe several ndividuals orming group,making collective ecision.None ofthe rgumentsgiven ere pply o the ase oftwo, rmore, ecisionmakers hodo nothave common urpose,ormay venbe inconflict. his s an mportantimitationn maximized xpected tility.One topic hat tatisticiansaveoften onsideredheir wn, t east ince hebrilliant ork fFisher 1935), is the design f experiments.his s a decision roblem nd fits eatly nto heprinciplesust enunciated. et e be a member f a class of possible xperimentsromwhich nemust e selected. et x denote ata hatmight risefrom uch n experiment.he experimentationpresumably as some purpose, xpressed y the selection f a (terminal) ecisiond. As usual,denote heuncertainlement y 0. A similar nalysis pplieswhen he nferences for utureatay.) The final onsequence f experimentationndaction s e, x, d, 0) to which ou ttach utilityu(e, x, d, 0). The expected utility s

Ju(e,x, d, 0) p(O e, x, d) dO, (5)the uncertaintyeingconditional n all the other ngredients.he optimum ecision s thatdwhichmaximizes xpression5). Denote hemaximum alue so obtained y -u(e,x). The expec-tation f his s

Ju(e,x) p(x e) dx, (6)sincex is the nly ncertainlementtthis tage, heuncertaintyeing learly ependentntheexperiment. A finalmaximizationfexpression6) provideshe ptimumxperimentalesign.Noticethe simplicity f theprincipleshat re involvedhere,even though he technicalmanipulations aybe formidable.here s a temporalequence hat lternatesetweenakingnexpectationverthequantitieshat re uncertainndmaximizingverthedecisions hat reavailable. achuncertaintyust e evaluatedonditionallynall that s knownhen. heutilityis attachedothefinal utcome,therexpected) tilities,ike u(e,x) beingderived herefrom.Thisprovides formal rameworkor hedesign fexperiments.Itwould ppear o be a sensible riticismfthemethodustoutlined hatmany xperimentsrenot conducted ith terminal ecisionnmindbutmerelyogathernformationbout0. Thisaspect an be accommodatedy extendinghe nterpretationfa decision. nformationbout0depends nyour ncertaintybout expressed,s always, yprobability.o letthedecision betoselect he elevant ensity,erep(O e, x). A utility unctionanthen e constructed.ftent sreasonableosuppose additiventhe ense hat

u(e, x, d, 0) = u(e, x) + u(d, 0), (7)thefirstermnvolvingheexperimentalostandthe econd heterminalonsequences. oticethe onnectionetween (d, 0) and he coringules uggestednSection .Hereu(d, 0) may ethoughtf as a reward core ttachedodecision toannounce (O e, x) when heparameterasvalue00.Theusualmeasurefthe nformationrovided yp(O) is Shannon's,

J(O) og{p(0)} dO.The language fdecision nalysis as been usedby Neyman ndothersn connection ith


23/46

314 D. V.Lindleyhypothesisesting, here hey peak of thedecisions oaccept, nd to reject, hehypothesis .There are cases where cceptance nd rejection an legitimatelye thoughtf as action, swith herejectionf a batchof items. quallythere reother ases wherewe could calculatep(Hlx, K) as an nferenceboutH on datax andknowledge . The atter ormmay, s in the astparagraph,e thoughtfas a decision. othformsrevalid nduseful or ifferenturposes. urphilosophyccommodatesoth iews nd t s for ou oconsider ow o model he eality eforeyou.Animportanteature f theBayesian aradigms itsability o encompass wide variety fsituationssing few asicprinciples.Somewriters,n discussing ypothesisesting, ave rguedhat here remany ifferentases.Forexample, omemayreally nvolve ction; ome repurelynferential.ther ases havebeendescribed,nding p, as with ikelihood,n a plethora f situationsnd great omplexity.heBayesianview is that hese are all covered y the general rinciples nd that he differencesperceivedre differencesntheprobabilityndutilitytructures.ome folk ove complexityorthides nadequaciesnd even rrors.16. Likelihood principle again)In Section13itwas seen howthe ikelihoodrinciplesbasic for nference,etdenied ymanyfrequentistotions. he principleeases to applywhen xperimentalesign spart fthedecisionanalysis, ssentiallyecauseofthe ntegrationverx involved n expression6). At the nitialstage,whereyou are considering hich xperimento perform,he data,conditional n anyexperimentelected,s uncertain oryou. This uncertaintys expressed hrough(xle) and iseliminatedytheoperationfexpectationn expression6). In conductingn inference,r inmaking terminal ecision, ouknow hevalueofx, for hedata re available.Consequentlyt sunnecessaryo consider ther atavaluesandthe ikelihood s all that s needed.When t is aquestion fexperimentalesign, he dataare surely otavailable nd all possibilitiesmustbecontemplated.hiscontrastetween re- ndpost-datamphasizeshe mportancef thecon-ditions henyoufaceuncertainty.robabilitys a functionf two rguments,ot ne.Just owtheconsiderationfthe xperimentaninvolve neform f integrationverx usedbyfrequentists,amelyrrorates, an be seen s follows. enotebyd* e, x) that ecisionwhichmaximizes he xpected tility5). The expectationver , expression5), canthen e writtenJJ{e,x,d*(e, x), O}p{0 e, x,d*(e, x)} dOp(xl ) dx.Now

p{Ole, x,d*(e, x)} = p(Ole,x)since he ddition fd*,a functionf e andx,addsnofurtherondition.he atterrobabilitys

p(x e, 0) p(O e)/p(x e).Insertinghis alue nto he xpectationndreversinghe rdersf ntegration,ehaveJu{e,d*(e, x), 0} p(x e, 0)dxp(0 e)dOwhere he nnerntegral xposes hefrequentistntegrationverx. For a fixed xperiment,ndwith utilityhat oesnotdirectlynvolve ,the elevantntegrals

Ju{d*(e,x), 0} p(x e, 0) dx.


24/46

Philosophyf tatistics 315With wo decisions nd 0-1 utility, e immediatelyavef (x e, 0)dx over subset fsamplespaceand hefamiliarrrors f he wokinds.The occurrence f error ates eads tosome confusion ecause they re often reated s thequantitieso be controlled,ndthereforeccupy primaryositionndecision nalysis,whereasourprimaryonsiderationies n theutilitytructure.nce theutilitytructureasbeen mposed,the errorswill look after hemselves. owever, considerationf differentrrorsmay ead toundesirablehangesntheutilitytructure.he Bayesian iew s that heutilities,ot he rrors,arethe nvariantsf he nalysis. or xample,o design nexperimento achieve rescribedrrorratesmaybe incoherent.he prescriptionhould nstead pecify tilities.17. RiskRisk s a termwhichwe havenotused. t has beendefinedDuckworth,998) as 'thepotential orexposureouncertainvents,he ccurrencefwhichwouldhaveundesirableonsequences'. hedefinitionecognizes hetwo elementsnwhatwe have called decision nalysis,heuncertaintyand the utility,hough uckworth,n commonwithmost tatisticians,mphasized he oss, orundesirability,atherhan hegain, heutility.hechanges linguistic.isk s thereforeependenton two rgumentsnd ourfoundationalresentationnSection isdependentn the eparationfthemnuncertaintyndworth. et t s common,s Duckworthoes, oquote measure frisk sa single umber,o denyinghe eparation. hus herisk ssociatedwith 1000-mile lights 1.7insuitable nits. his s defensibleor hefollowingeason.The optimumecisionmaximizes xpected tility hich, or atax, s proportionalo

ju(d, 0) p(0) p(x 0) dOandmay ewrittens aweightedikelihood

jw(d, 0) p(x 0) dO,wherew(d, 0) = u(d, 0) p(O). Theanalysis,or iven ata, ndhencefor given ikelihood,oesnotdepend eparatelyntheutilityndprobability,he wocorner-stonesf thephilosophy,utonly n their roduct.oput t nanother ay,fyouwere owatch coherenterson cting asdistinctromxpressingisthoughts) ouwouldnot, n thebasis ofthe bservedctions, e abletoseparatehe wo lements; nly heweightunction ighte determined.Nevertheless here re several reasons forseparating tility romprobability.he mostimportants theneedfor nference,.e. for sound ppreciationftheworldwithout eferenceoaction.Thephilosophyaysthat his s hadthrough our robabilitytructureor heworld.ninference, anipulationsake place entirely ithin he probabilityalculus,which thereforebecomes eparated rom tility.here repeoplewhoargue hatnference,nthe form fpurescience,s unsatisfactoryhen solated romtsapplicationsntechnology. hat s undoubtedlyimportants that nferencehould e ina suitable orm or ecision-making,ot n activityhat sisolatedfrom pplication.We have seen howBayesian nferences perfectlydaptedfor hispurpose. t will be seen in Section 19 how someaspectsof the aw separatenferenceromdecision.Another easonfor heseparationies in thedesirabilityfcommunicationetween eople,between ifferentyous'. Take theexampleof the 1000-mile lightited above. Partof thecalculation ests n the bserved ccident ate or ircraft. notherart ests nthe onsequencesof theflight.oumayreactdifferentlyothese wo elements. orexample,t s known hat or


25/46

316 D. V.Lindleyelderly eople heres an ncreased isk fcirculatoryroblems uetositting or oursncrampedseats, ndthereforeoumay valuate our ccident ate ifferentlyrom hat uggested urely ytheaccident atefor ircraft.n contrast, healthy, iddle-agedxecutive, ravellingnmorecomfortnfirstlass,may ccept he ccident tatisticsuthavea differenttility ecauseof theimportancef themeeting o whichhe is bound.Theseconsiderationsuggest hat he ccidentrate nd consequences f n accident e kept eparate ecauseyoumaybe able to use one elementbutnot he ther, hereas heweight unctionlonewould e more ifficulto use.18. ScienceKarl Pearson aid 'The unity f all science consists lone in itsmethod, ot n its material'(Pearson, 892). It is nottrue o say that hysics s sciencewhereas iteratures not.There retimeswhen physicist akes leap ofthe maginationike n artist. nalyses fword ounts anhelp o dentifyhe uthor f n anonymousiece of iterature.cientific ethodscertainly uchmore mportantn physics han n iterature,ut thas thepotentialityo be used n anydiscipline.Ofwhat hen oesthemethodonsist? here s an enormousiteratureevoted o answeringthisquestion ndit is presumptuousfmeto claim to have theanswer. ut I do believethatstatisticians,ntheir eep study f thecollection nd analyses f datahave,perhaps nwittingly,uncovered he answer nd it lies in thephilosophy resented ere.Experimentation,ith tsproductionf data, s an essential ngredientf scientific ethod,o the connection etweenstatisticsnd science s notsurprising.n thisview, he scientific ethod onsistsnexpressingyour iewofyour ncertain orldnterms fprobability,erformingxperimentso obtain ata,andusing hatdata toupdateyourprobabilitynd henceyourview of theworld.Althoughheemphasisn this pdatings ordinarilyutonBayes, ffectivelyheproduct ule, he liminationof theubiquitous uisance arametersythe ddition ule2 is also important.s we haveseen,thedesign fthe xperiments also amenable o statisticalreatment.cientific ethodonsistsfa sequence lternatingetween easoningndexperimentation.s explainedn Section , eachscientists a 'you'with heir wnbeliefswhich rebroughtntoharmonyhroughhe ccumula-tion fdata. t sthis onsensushats objective cience.Objections avebeenmadetothis imple iewonthegroundshat cientists o not ct n thewaydescribednthe astparagraph. hey vendotailareasignificanceests. heresponseo theobjections that urphilosophys normative,otdescriptive.t is notthe ntentiono describehow cientistsehavebuthowtheywouldwish o behave fonly hey newhow.Theprobabilitycalculusprovides he how'. An impedimentffectinghow' is the ack of goodmethods fassessing robabilities henno exchangeabilityssumptions availableto guide you.This isordinarilyescribeds determiningour rior,ut nrealityt swider han hat. ome attacks nscience retrulyttacksn how cientistsehave-onthedescriptivespect.Oftenhey revalid.Such attackswould become ess cogent ftheydealt with he normativespect.Scientists rehuman.Real scientistsre affectedyextraneous onditions. ne wouldhopethat scientistworkingor multinationalompanynd anothermployed yan environmentalgencydifferonlyntheir robabilitiesnd wouldupdate ccordingly.nesuspectshat therssues ntervene.It smyhope hat Bayesian pproach ouldhelp oexpose nybiasesorfallaciesneither ftheprotagonists'rguments.19. Criminal awThere re tworeasons orncludinghis ection n criminalaw: first ecauseofmy wn nterestinforensiccience; econdbecauseof theconvictionhat his nterest asengenderedhat ome


26/46

Philosophyf tatistics 317importantspects fthe aware amenable o the cientific ethod s describednthe ast ection.Theseaspects oncern he trialprocess,where here s uncertaintyboutthedefendant'suilt,uncertaintyhat s subsequentlyempered y data, ntheformf evidence, opefullyo reachconsensus bout heguilt. learlyhis itsnto heparadigm eveloped ere. awyers onothavea monopolyn thediscovery f the ruth;cientists avebeen doing tsuccessfullyor enturies.There reaspects f the aw, ikethewritingfa law, owhich he cientific ethod as little ocontribute.owever,he ourts re notustconcerned ith uilt; hey eedto pass sentence. helaw hasseparated hese wo functions,ust as we have.They an be recognized s inferencenddecision-makingespectively.The defendantn a court f aw seitherruly uilty or notguilty G. Theguiltsuncertainand so should e described ya probability(G). (Thebackground nowledges omittedromthenotation.) ata,inthe form f evidence , are produced ndtheprobabilitypdated. incethere reonly wopossibilities, or - G, t s convenientoworkntermsfodds on),

o(G) p(G)/p(G),whenBayes's heoremeadso(GIE) p(E IG o(G)p(El G)oG

involvingmultiplicationf the original dds by the ikelihood atio.Evidenceoften nvolvesnuisance arametersut, nprinciple,hese an be eliminatedn theusual waybythe additionrule.Theywilloften nternto (E- G) because heremight e severalways nwhich he rimecouldhavebeencommitted,ther han ythedefendant. s the rial roceeds, urthervidencesintroducednd successivemultiplicationsy likelihood atiosdeterminehefinal dds.A dif-ficultyere s that uccessive iecesofevidencemaynotbe independent,ither ivenG orgivenrG.So far hismethod asmainly eenusedsuccessfullyor cientificvidence,ike bloodstainsand DNA (AitkenndStoney,991). tsapplicabilityngeneral epends nsatisfactoryethodsofprobabilityssessment.t hasthepotentialdvantagefhelpinghe ourt ocombine isparatetypes fevidence or, s remarkedn Section3, theprincipalmerit f measurementies in itsabilityo meld everal ncertaintiesnto ne.The law agrees with he philosophyn separatingnference rom ecision. t even allowsdifferentvidence o be admittednto he woprocesses. orexample, revious onvictions aybe used in sentencingdecision)butnotalways n assessingguilt inference).xpected tilityanalysisncludes theoremo the ffecthat ost-freenformations always xpectedo ncreasetheutility.hissuggestshat heonly eason ornot dmittingvidence hould e ongroundsfcost Eggleston, 983).Thepart fthe rial rocess hat,tpresent,esultsntheudgment uilty,r notguilty,hould,in ourview,be replaced ythe calculation f odds o(GIE), whereE is now thetotalityf alladmittedvidence.On thisview, he ury houldnotmakea firmtatementfguilt, rnot,butstate heir inal dds, rprobability,fguilt.At east his rovides more lexiblend nformativecommunication. ore mportantly,tprovideshe udgewith he nformationhathe needsforsentencing.fd is a possible ecision,bout aolora fine,hen he xpected tilityfd is

u(d, G) p(G IE) + u(d, -G) p(-G IE).Theoptimumentences that whichmaximizes his xpectation.he utilities erewillreflectsociety'svaluation fthemerits f differententences or heguilty erson,nd the eriousness


27/46

318 D. V.Lindleyof false mprisonment.e are a longwayfrom he mplementationfthese deas butevennowthey an guideus nto ensible roceduresnd avoid ncoherentnes.20. ConclusionsThephilosophyf statistics resented erehas three undamentalenets: irst,hat ncertaintyshould edescribedyprobabilities;econd, hat onsequenceshould ave heirmeritsescribedbyutilities;hird,hat he ptimum ecision ombines heprobabilitiesndutilitiesycalculatingexpected tilitynd henmaximizinghat.fthese re ccepted,hen he irstask f statisticiansto develop (probability) odel o embrace he lient'snterestsnduncertainties.t will ncludethedataand anyparametershat re udgednecessary. nceaccomplished,hemechanics f thecalculus ake ver ndthe equirednferencesmade. fdecisionsre nvolved,hemodelneeds obe extendedo nclude tilities,ollowedy nother echanicalperationfmaximizingxpectedutility.ne attractiveeatures that hewhole rocedures welldefinednd heres ittle eedfor dhoc assumptions. here s, however, considerableeed for pproximation.o carry ut thisscheme or he argeworld s impossible.t is essential o use a smallworld,which ntroducessimplificationut often auses distortion.ven themechanics f calculation eed numericalapproximations.oth hese ssues have been considerednthe iterature,hetherrequentistrBayesian, nd substantial rogress as been made. Where a real difficultyrisesis in theconstructionf themodel.Manyvaluable echniques ave been ntroducedut,because of thefrequentistmphasisn past work, here s a real gap in our appreciationf how to assessprobabilities-of owtoexpress uruncertaintiesntherequisiteorm.Myview s that hemostimportanttatisticalesearch opic s we enter henewmillenniumsthedevelopmentfsensiblemethods fprobabilityssessment. his will require o-operation ithnumeratexperimentalpsychologistsndmuch xperimentalork.A colleague ut tneatly,hough ith ome xaggera-tion:There renoproblemseftnstatisticsxcepthe ssessmentfprobability'.t s curious hatthe ypical xpertnprobabilitynows othingbout,ndhasno nterestn, ssessment.The adoptionof the positionoutlinedn thispaper would result n a widening f thestatistician'semit oinclude ecision-making,s wellas datacollection,model onstructionndinference. et it also involves restrictionn their ctivityhat has not been adequatelyrecognized.tatisticiansre notmastersn their wn house. Their ask s tohelpthe client ohandle theuncertaintyhat hey ncounter. he 'you' of the analysis s the client,not thestatistician.ur ournals, ndperhaps urpractice, ave been too divorced rom heclient'srequirements.n this havebeen as guilty s any.But at least the theoreticianas developedmethods. our ask s toput hem ogooduse.ReferencesAitken, . G. G. and Stoney,. A. (1991) The Use of tatisticsnForensic cience.Chichester: orwood.Bartholomew,. J. 1988) Probability,tatisticsnd heologywith iscussion). R. Statist. oc.A, 151,137-178.Berger, . . andDelampady, . (1987) Testing recise ypotheseswith iscussion). tatist. ci., 2, 317-352.Berger, . . andWolpert, . L. (1988) TheLikelihood rinciple. ayward:nstitutefMathematicaltatistics.Bemardo,J.M. (1999) Nestedhypothesisesting: he Bayesian eferenceriterion.n BayesianStatistics (eds J.M.Bemardo, . . Berger, . P.Dawid andA. F M. Smith). xford: larendon.Bemardo, .M., Berger, .O.,Dawid,A. P. andSmith, . F M. (eds) 1999) Bayesian tatistics. Oxford: larendon.Bemardo, .M. andSmith, . F M. (1994) BayesianTheory. hichester: iley.Box, G. E. P. 1980) Samplingnd Bayes' inferencen scientific odellingwith iscussion). R. Statist. oc. A, 143,383-430.Box, G. E. P.andCox,D. R. (1964)An analysis ftransformationswith iscussion). R. Statist. oc.B, 26, 211-252.Dawid, A. P., Stone,M. and Zidek,J.V (1973) Marginalizationaradoxes n Bayesian nd structuralnferencewithdiscussion). R. Statist. oc. B, 35, 189-233.


28/46

Philosophy fStatistics 319DeGroot,M. H. (1970) Optimal tatistical ecisions.NewYork:McGraw-Hill.Draper, . (1995) Assessmentndpropagationfmodel ncertaintywith iscussion). R. Statist. oc. B, 57, 45-98.Duckworth,. 1998) The quantificationfrisk.RSSNews, 6,no.2, 10-12.Edwards,W. L., Lindman, . andSavage,L. J. 1963) Bayesian tatisticalnferenceor sychologicalesearch. sychol.Rev., 0, 193-242.Eggleston, . (1983) Evidence, roof ndProbability.ondon:WeidenfeldndNicolson.deFinetti, . (1974) Theory fProbability,ol. 1.Chichester: iley.(1975) Theory fProbability,ol. 2. Chichester: iley.Fisher, . A. (1935) TheDesignofExperiments.dinburgh:liver ndBoyd.Healy,M. J.R. (1969) Rao's paradox oncerning ultivariateests f ignificance.iometrics,5, 411 413.Jeffreys,. (1961) Theory fProbability.xford: larendon.Lehmann,. L. (1983) Theory fPoint stimation. ew York:Wiley.(1986) Testingtatistical ypotheses. ew York:Wiley.O'Hagan,A. (1995)Fractionalayesfactors ormodel omparisonwith iscussion). R. Statist. oc. B,57,99-138.Onions, . T. ed.) (1956) The horternglish ictionary.xford: larendon.Pearson, . (1892) TheGrammarf cience.London:Black.Ramsey, . P. (1926) Truth ndprobability.n TheFoundations fMathematicsnd Other ogical Essays (ed. R. B.Braithwaite),p. 156-198. London:KeganPaul.Savage, . J. 1954) TheFoundationsf tatistics.ew York:Wiley.(1977) The shiftingoundationsf statistics.n Logic, Laws and Life:Some PhilosophicalComplicationsed.R. G. Colodny), p.3-18. Pittsburgh:ittsburghniversityress.Stein,C. (1956) Inadmissibilityf the usual estimationf themeanofa multivariateormal istribution.n Proc. 3rdBerkeleyymp.Mathematicaltatisticsnd ProbabilityedsJ.NeymanndE. L. Scott), ol. 1,pp. 197-206.Berkeley:UniversityfCalifornia ress.Walley, . 1991) Statistical easoningwithmprecise robabilities.ondon:Chapman ndHall.

Comments on the paper byLindleyPeter Armitage Wallingford)Dennis Lindleyhaswritteno frequently,nd so persuasively,bout heprinciples f Bayesian tatistics,thatwe scarcely xpectto findnew insightsnyet another uch paper.The present aper showshowwrong ucha prior udgmentwouldbe. Lindley's oncern s with heverynature f statistics,ndhisargument nfolds learly, eamlessly ndrelentlessly.hose ofuswhocannot ccompany im o the ndof his oumeymust onsider ery arefully hereweneed todismount; therwise e shallfind urselvesunwittinglytthebusterminus, ithout returnicket.I wrote thoseofus' because theremust emanywho, ikeme, sympathize ithmuch f theBayesianapproach utareunwillingodiscard frequentistradition hich ppears ohave served hemwell. Itis worth ryingoenquirewhy his houldbe so. One possibility,fcourse, s that ur reluctances, atleast n part, manifestationf inertia, emonstratinglackofcourageorunderstanding.must eavethat or thers o udge. I think, hough,hat here re sounder easonsforwithholding ull upport ortheBayesianposition. indley nd cametostatistics uring he 1940s, at a timewhen he subjectwasdominated y the Fisherianrevolution. uringthe 19thcenturynverseprobability ad co-existeduneasilywithfrequentist ethods ytheuse offlat riors, tandardrrors nd normal pproximations,results eing nterpretableyeithermodeofreasoning, lbeitwith ccasional ack ofclarity. isherhad,it seemed, leared he irby disposing ftheneedfor nverse robability.hilosophical isputes, uch asthosewithNeyman nd E. S. Pearson, ookplace within hefrequentistchool, lthough effreysndafew ther ioneersmaintained nddeveloped heLaplace-Bayes framework.omany f us enteringhefield t that ime twouldhave seemedbizarre o overtum uch a powerful odyofideas. It is greatlyothecredit f Lindley, nd of a fewof hiscontemporariesike Good andSavage,that hey ecognized hepossibility hat, s theymight ave put t, heEmperor adnoclothes.The greatmerit f the Fisherian evolution,partfromhesheerrichness f theapplicablemethods,was the abilityto summarize, nd to draw conclusionsfrom, xperimental nd observational atawithout eference o prior eliefs.An experimentalcientist eedstoreport is or herfindings,ndtostatea rangeof possiblehypotheseswithwhich thesefindingsre consistent. he scientistwill un-doubtedly aveprejudices ndhunches, utthereportingfthese should not be a primaryim of theinvestigation.onsider, or nstance, ne ofthemajor chievementsfmedical tatisticsnthe ast half-century-the irsttudy fDoll and Hill (1950) on smoking nd lung cancer.They certainly ad priorhunches, .g. that irpollutionwas more ikely han moking o cause lungcancer,but t would haveserved no purposeto quantify hesebeliefsand to enter hem ntothe calculations hrough ayes's


29/46

the philosophy of statistics

Documents