The Philosophy of Statistics Author(s): Dennis V. Lindley Reviewed work(s): Source: Journal of the Royal Statistical Society. Series D (The Statistician), Vol. 49, No. 3 (2000), pp. 293-337 Published by: Wiley-Blackwell for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2681060 . Accessed: 05/09/2012 14:41 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . 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]. Wiley-Blackwell and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series D (The Statistician). http://www.jstor.org The Statistician(2000) 49, Part3, pp. 293-337 The philosophy of statistics Dennis V. Lindley Minehead,UK [Received June 1999] Summary. This paper puts forwardan overallview of statistics.It is argued that statisticsis the studyof uncertainty.The many demonstrationsthat uncertaintiescan only combine accordingto the rules of the probabilitycalculus are summarized.The conclusion is thatstatistical inference is firmlybased on probabilityalone. Progress is thereforedependent on the constructionof a probabilitymodel; methods for doing this are considered. It is argued that the probabilitiesare personal. The roles of likelihoodand exchangeabilityare explained. Inferenceis onlyof value ifit can be used, so the extensionto decision analysis, incorporatingutility, is relatedto riskand to the use of statisticsin science and law. The paper has been writtenin the hope thatit will be intelligible to all who are interestedin statistics. Keywords:Conglomerability; Data analysis; Decision analysis; Exchangeability;Law; Likelihood; Models; Personal probability;Risk; Scientificmethod; Utility 1. Introduction Insteadof discussinga specificproblem, this paper providesan overviewwithin which most statisticalissues can be considered.'Philosophy' in thetitle is used in thesense of 'The studyof the generalprinciples of some particularbranch of knowledge,experience or activity'(Onions, 1956).The wordhas recentlyacquired a reputationfor being concerned solely with abstract issues, divorcedfrom reality. My intentionhere is to avoidexcessive abstraction and to deal withpractical mattersconcerned with our subject. If thepractitioner who readsthis paper does notfeel that the studyhas benefitedthem, my writing will havefailed in one of itsendeavours. The papertries to developa wayof lookingat statisticsthat will help us, as statisticians,to developbetter a sound approachto anyproblem which we mightencounter. Technical matters have largely been avoided, notbecause theyare not important,but in theinterests of focusingon a clear understandingof how a statisticalsituation can be studied.At someplaces, matters of detailhave been omittedto highlightthe key idea. For example,probability densities have been used withoutan explicit mentionof the dominating measure to which they refer. The paperbegins by recognizingthat statistical issues concernuncertainty, going on to argue thatuncertainty can only be measuredby probability.This conclusionenables a systematic accountof inference,based on theprobability calculus, to be developed,which is shownto be differentfrom some conventionalaccounts. The likelihoodprinciple follows from the basic role playedby probability.The role of data analysisin theconstruction of probabilitymodels and the natureof modelsare nextdiscussed. The developmentleads to a methodof makingdecisions and the natureof riskis considered.Scientific method and its applicationto some legal issues are explainedwithin the probabilistic framework. The conclusionis thatwe have herea satisfactory generalset of statisticalprocedures whose implementation should improve statistical practice. Addressfor correspondence: Dennis V Lindley,"Woodstock", Quay Lane, Minehead, Somerset, TA24 5QU, UK. E-mail:[email protected] ? 2000 Royal StatisticalSociety 0039-0526/00/49293 294 D. V. Lindley The philosophyhere presented places moreemphasis on modelconstruction than on formal inference.In thisit agreeswith much recent opinion. A reasonfor this change of emphasisis that formalinference is a systematicprocedure within the calculus of probability. Model construction, bycontrast, cannot be so systematic. The paperarose out of myexperiences at theSixth Valencia Conference on BayesianStatistics, heldin June1998 (Bemardoet al., 1999). AlthoughI was impressedby theoverall quality of the papersand thesubstantial advances made, many participants did notseem to me fullyto appre- ciatethe Bayesian philosophy. This paper is an attemptto describemy version of thatphilosophy. It is a reflectionof 50 years'statistical experience and a personalchange from a frequentist, throughobjective Bayes, to the subjectiveattitude presented here. No attempthas been made to analysein detailalternative philosophies, only to indicatewhere their conclusions differ from those developedhere and to contrastthe resulting practical methods. 2. Statistics To discuss the philosophyof statistics,it is necessaryto be reasonablyclear what it is the philosophyof, not in thesense of a precisedefinition, so thatthis is 'in', thatis 'out', butmerely to be able to perceiveits outlines. The suggestionhere is thatstatistics is thestudy of uncertainty (Savage, 1977): thatstatisticians are expertsin handlinguncertainty. They have developedtools, like standarderrors and significancelevels, that measure the uncertainties that we mightreason- ablyfeel. A checkof howwell this description of oursubject agrees with what we actuallydo can be performedby lookingat thefour series of journals published by the Royal Statistical Society in 1997. These embraceissues as diverseas social accountingand stablelaws. Apartfrom a few exceptions,like thealgorithms section (which has subsequentlybeen abandoned)and a paperon education,all thepapers deal eitherdirectly with uncertainty or withfeatures, like stablelaws, whicharise in problemsthat exhibit uncertainty. Support for this view of oursubject is provided by the fact that statisticsplays a greaterrole in topics thathave variability,giving rise to uncertainty,as an essentialingredient, than in moreprecise subjects. Agriculture, for example, enjoysa close associationwith statistics, whereas physics does not. Notice thatit is onlythe manipulationof uncertaintythat interests us. We are not concernedwith the matterthat is uncertain.Thus we do not studythe mechanismof rain;only whether it will rain.This places statisticsin a curioussituation in thatwe are,as practitioners,dependent on others.The forecastof rainwill be dependenton bothmeteorologist and statistician.Only as theoreticianscan we exist alone.Even therewe sufferif we remaintoo divorcedfrom the science. The term'client' will be used in referenceto theperson, e.g. scientistor lawyer,who encountersuncertainty in theirfield of study. The philosophicalposition adopted here is thatstatistics is essentiallythe study of uncertainty and thatthe statistician'srole is to assist workersin otherfields, the clients,who encounter uncertaintyin theirwork. In practice,there is a restrictionin thatstatistics is ordinarilyassociated withdata; and it is thelink between the uncertainty, or variability, in thedata and that in thetopic itselfthat has occupiedstatisticians. Some writerseven restrictthe data to be frequencydata, capable of near-identicalrepetition. Uncertainty, away fromdata, has rarelybeen of statistical interest.Statisticians do nothave a monopolyof studiesof uncertainty.Probabilists discuss how randomnessin one partof a systemaffects other parts. Thus themodel fora stochasticprocess providespredictions about the data that the process will provide. The passagefrom process to data is clear;it is whenwe attempta reversaland go fromdata to processthat difficulties appear. This paperis mainlydevoted to thislast phase, commonly called inference, and theaction that it might generate. Philosophyof Statistics 295 Notice thatuncertainty is everywhere,not just in science or even in data. It providesa motivationfor some aspects of theology(Bartholomew, 1988). Therefore,the recognitionof statisticsas uncertaintywould imply an extensiverole forstatisticians. If a philosophicalposition can be developedthat embraces all uncertainty,it will providean importantadvance in our understandingofthe world. At the moment it would be presumptiveto claimso much. 3. Uncertainty Acceptancethat statistics is thestudy of uncertaintyimplies that it is necessaryto investigatethe phenomenon.A scientificapproach would mean the measurementof uncertainty;for, to follow Kelvin,it is onlyby associatingnumbers with any scientificconcept that the conceptcan be properlyunderstood. The reasonfor measurement is notjust to makemore precise the notion that we are moreuncertain about the stock-market than about the sun rising tomorrow, but to be able to combineuncertainties. Only exceptionally is thereone elementof uncertaintyin a problem; morerealistically there are several.In thecollection of data,there is uncertaintyin thesampling unit,and thenin thenumber reported in thesampling. In an archetypalstatistical problem, there is uncertaintyin bothdata and