Table Of Content

StatisticalScience 2011,Vol.26,No.3,329–331 DOI:10.1214/11-STS352REJ MainarticleDOI:10.1214/11-STS352 R(cid:13)c InsetitujteoofMianthemdaticealSrtatistics,2011 D. A. S. Fraser 1. INTRODUCTION A statistical model differs from a deterministic model in having added probability structure that Verysincerethankstothediscussantsforchoosing describes the variability typically present in most 2 to enter a virtual minefield of disagreement in the applications. So, in an application with a statistical 1 development history of statistics. For we just need 0 model and related data it would then seem quite torecall theremarkthatfiducialisFisher’s“biggest 2 natural that that variability would enter the con- blunder” and place it alongside the fact that fidu- n clusions concerning the unknownsin an application: cial was the initial step toward confidence, which a what do I know deterministically, and what do I J arguably is the most substantive ingredient in mod- know probabilistically? 3 ern model-based theory: the two differ in minor de- And that is what Bayes proposed in 1763: prob- velopmental detail, with fiducialoffering a probabil- ] ability statements concerning the unknowns of an E ity distribution as does Bayes and with confidence investigation. Many have had doubts and said there M offering just probabilities for intervals and special was no merit in the proposal; and many have ac- regions. Statistics has spent far more time attack- . cededandbecamestrongbelievers. AndthenFisher t ing incremental steps than it has seeking insightful a (1930) also offered probabilities concerning the un- t resolutions. s knowns of an investigation, but by a different argu- [ Asamoderndisciplinestatistics hasinheritedtwo ment,andtheturffightbegan!Bayes hadhesitantly prominentapproachestotheanalysisofmodelswith 1 examined a special problem and added a random v data; of course such is not all of statistics but is generator for the unknown parameter, and Fisher 1 acriticalportionthatinfluencesthedisciplinewidely. 1 had worked more generally and used just the ran- How can a discipline, central to science and to crit- 6 domness that had generated the data itself. 0 ical thinking, have two methodologies, two logics, But then a third person, Lindley (1958), from the . two approaches that frequently give substantially 1 same country said that the second person, Fisher, 0 different answers to the same problems. Any astute couldn’t use the term probability for the unknowns 2 personfromoutsidewouldsay,“Whydon’ttheyput 1 in an investigation, as the term was already taken their house in order?” And any serious mathemati- : by the first person, Bayes. And strangely the disci- v cian would surely ask how you could use a lemma i plinecomplied!Decadeswentbyandanecdoteswere X with one premise missing by making up an ingredi- traded and things were often vitriolic. ent and thinking that the conclusions of the lemma r a were still available. Of course, the two approaches 2. WHAT DOES THE ORACLE SAY? have been around since 1763 and 1930 with regular disagreement and yet no sense of urgency to clarify Consider some regular statistical model f(y;θ), the conflicts. And now even a tired discipline can togetherwithalowerβ-confidenceboundθˆ (y),and β just ask, “Who wants to face those old questions?”: also alower β-posterior boundθ˜(y) based on a prior a fully understandablereaction! But is complacency π(θ): What does the oracle see concerning the us- in the face of contradiction acceptable for a central age of these bounds? He can investigate any long discipline of science? sequence of usages of the model, and He would have availablethedatavaluesy andofcoursethepreced- i D. A. S. Fraser is Professor, Department of Statistics, ing parameter values θ that producedthe y values; i i University of Toronto, Toronto, Ontario, Canada M5S He would thus have access to {(θ ,y ):i=1,2,...}. i i 3G3 e-mail: [email protected]. Firstconsiderthelowerconfidencebound.Theor- acle knowswhether ornotthe θ is intheconfidence This is an electronic reprint of the original article i interval (θˆ (y ),∞), and He can examine the long- published by the Institute of Mathematical Statistics in β i Statistical Science, 2011, Vol. 26, No. 3, 329–331. This run proportion of true statements among the asser- reprint differs from the original in pagination and tionsthatθi isintheconfidenceinterval(θˆβ(yi),∞), typographic detail. and He can see whether the confidence claim of a β- 1 2 D. A.S. FRASER proportion true is correct. In agreement with the by other methods of analysis. Ifone faces a probabi- mathematicsofconfidence,thatproportionisjustβ. lity-typeclaim,itisfairenoughtosimulateandeval- Now consider the lower posterior bound. The or- uatetheclaim,andthatiswhatcoverageprobability acle knows whether θ is in the posterior interval is all about, as the invincible Oracle well knows. i (θ˜ (y ),∞), and He can examine the long-run pro- The marginalization paradoxes do appear in the β i portion of true statements that θ is in the poste- literature but are widely neglected and not “exten- i rior interval (θ˜ (y ),∞). Now suppose the long-run sively discussed” as Christian suggests. They apply β i pattern of θ values just happened to correspond to to any proposal for a distribution to describe an un- i the pattern π(θ); then in full agreement with the known vector parameter, whether obtained by the mathematics of the Bayes calculation, the Oracle Bayes inversion of a density or the frequentist in- would see that long-run proportion of true state- version of a pivot, such as fiducial, confidence struc- mentsamongassertionsthatθ isintheposteriorin- tural or other. There is an immutable contradiction i terval (θ˜ (y ),∞) wascorrect, was justthestated β. built into the hope to describe a vector parameter β i But what if the long-run pattern of θ values was by a distribution. Curvature of an interest parame- i different from the introduced π(θ) pattern? Then in ter has emerged as the critical source for this con- wide generality the long-run proportion of true sta- tradiction. Take a bivariate parameter, a data point tements among assertions that θ is in the posterior and an interest parameter value: if the parameter i interval (θ˜ (y ),∞) would not be β! In other words, is linear, the confidence and the Bayes values are β i the confidence procedure is always right, and the equal; if then parameter curvature is introduced, we Bayes procedure is typically wrong, unless the prior have that the confidence value and the Bayes value was guessed correctly. Seems like a poor trade-off! change in opposite directions! One has the cover- Now consider further what a prior actually does age property and the “other” acquires bias at twice in producing parameter bounds or quantiles that the rate of the departure from linearity. And the are different from the confidence bound. From an “other”usesthenameprobabilitywithanassertive- asymptotic viewpoint a prior can be expanded as nesscomingfromtheuseoftheprobabilitycalculus, exp(aθ/n1/2 + cθ2/n) to the third order, as men- conveniently overlooking that an artifact was intro- tioned but not pursued in Section 6(iv). This pro- duced in place of the input needed for the validity vides a direct displacement of the confidence bound of the probability calculus for the application. in standardized units and produces an O(1)-shift MaybeitistimetoaddressthePandora’sboxand away from the claimed β value, either up or down check for a Madoff pyramid: too good to be true. depending on the sign of a! Hardly an argument for Larry Wasserman using the Bayes procedure unless there was some very urgent need for a quick and dirty calculation. Larry presents a pragmatic view of the Bayes approach, acknowledging its rich flexibility butrecom- 3. RESPONSE TO THE DISCUSSANTS mending coverage cautions. His five examples are most welcome concerning the wider spheres of ap- Christian Robert plication and he is to be complemented on the skill- Christian presents a very committed Bayes view- ful innovations. I do quarrel, however, with his rein- pointandquitecorrectly admonishesmefornotdis- forcement of personality cults in statistics. It seems tinguishing what Thomas Bayes did and what has that statistics has suffered greatly from this exter- followed in the same theme. But going beyond the nalization of the scientific method, as if there were minor detail, Bayes added a distribution for a pa- differentflavorsofscientificthinkingandmathemat- rameter, a distribution that was not part of the bi- ical logic and that these might gain concreteness nomial example under consideration and then used when personalized. thatdistributionfor probability analysis. Andmuch Kesar Singh and Minge Xie ofmodernBayesianstatisticsdoespreciselythat:in- troduces an artifact distribution for expediency or Confidence for estimation and exploration? It is convenience and then works reassuringly within ac- deeply unfortunate that statistics chooses at many cepted probability calculus. Indeed, this is the pri- steps to malign its major innovators, for example, mary theme of the article: adding something arbi- Fisher with his “biggest blunder” as a referent for trary gives something arbitrary no matter how at- the initiative that gave us confidence. What Fisher tractive the material labeled probability might or didn’tdowaspresenthismajorinnovationsinafully might not be, or no matter what might be available packaged form ready to withstandafew centuries of 3 REJOINDER challenges and modification: What? We still have to different sources can be reported separately, with do a little bit of thinking! Tough! He clearly must combinationnot byprinciple.Combininglikelihoods generously have expected others to have his insight is a consequence of combining models, typically fol- and wisdom! lowing from independence; the Bayes claim that it Fiducial, confidence,structural or other? Itis just comes from the use of the Bayes argument is after pivot inversion with variation in context, conditions the fact and disingenuous. Inverting a density and or interpretation: the big risk was described by Da- inverting a pivot are different except in the linear wid, Stone and Zidek (1973) and curvature is now case, but the first can sometimes approximate the identified as the prime cause. To have different na- second. mestofinetunefordifferentapplicationsordifferent The question was asked: “Is Bayes posterior just explorationswouldseemtotakeemphasisawayfrom quickanddirtyconfidence?”Andthecasewasmade the proper calibration of the tool, as the primary for “Yes”: Bayes posterior is just quick and dirty concern for most applications. confidence: quick in the sense of easier than using Statistics routinelycombineslikelihoodsasappro- quantiles to determine how θ affects data; and ap- priate, so it is not correct to attribute this to Baye- proximate in the sense of a wide spread need to use sian learning; perhaps the central sectors of statis- approximation methods. tics were just slow to glamorize the good things in Not everyone liked the blunt question. One dis- their statistical modeling. Putting a prior on a like- cussion expresses discomfort with such a direct con- lihood is a different operation downstream from as- frontation to the Bayes approach; one discussion sembling the likelihood in the relevant broader con- adds additional support examples; and the two re- text, although it doesseem convenient for theBayes mainingdiscussionsspeakmoretomethodsandmod- approachtoco-optitastheirowncontributionwhen ificationsofconfidencedistributions,overlookingthe it was somewhat neglected by the “others.” risks. But no one argued that the use of the condi- tional probability lemma with an imaginary input Tong Zhang had powers beyond confidence, supernatural pow- Where does the pivot come from? Fisher’s de- ers. velopment of confidence or whatever attracted the mathematicians’criticisms,mostlybecauseitwasn’t ACKNOWLEDGMENT proposedinafullydevelopedform.Itwasthenshred- ded, fully ignoring the emerging recognition of its The author acknowledges the support of the Nat- innovative genius. Certainly the need to clarify the ural Sciences and Engineering Research Council of origin of the key ingredient, “the pivot,” is of fun- Canada. damental importance, as Tong suggests: using all the data in an appropriately balanced way, respect- ing continuity and parameter direction from data, REFERENCES and more. Whether these should be bundled under Dawid,A.P.,Stone,M.andZidek,J.V.(1973).Marginal- a term optimality may be questionable, but doesn’t ization paradoxes in Bayesian and structural inference diminish the importance of the individual criteria; (with discussion). J. Roy. Statist. Soc. Ser. B 35 189–233. for some recent emphasis on continuity see Fraser, MR0365805 Fraser and Staicu (2010). Fisher, R. A.(1930).Inverseprobability.Proc. Camb. Phil. Soc. 26 528–535. Fraser,A.M.,Fraser,D.A.S.andStaicu,A.M.(2010). 4. SOME CONCLUDING INVOCATIVE Secondorderancillary:Adifferentialviewwithcontinuity. REMARKS Bernoulli 16 1208–1223. Lindley,D.V.(1958).FiducialdistributionsandBayes’the- An inference distribution for a vector parameter orem.J. Roy. Statist. Soc. Ser. B20102–107. MR0095550 is inherently a contradiction. Information from two