Who Should Be Afraid of the Jeffreys-Lindley Paradox?

21 Pages·2013·0.51 MB·English

Who Should Be Afraid of the
Jeffreys-Lindley Paradox?
Aris Spanos*y
ThearticlerevisitsthelargenðsamplesizeÞproblemasitrelatestotheJeffreys-Lindley
paradoxtocomparethefrequentist,Bayesian,andlikelihoodistapproachestoinference
andevidence.Itisarguedthatwhatisfallaciousistointerpretarejectionof H aspro-
0
vidingthesameevidenceforaparticularalternativeH ,irrespectiveofn;thisisanex-
1
ampleofthefallacyofrejection.Moreover,theBayesianandlikelihoodistapproaches
areshowntobesusceptibletothefallacyofacceptance.Thekeydifferenceisthatinfre-
quentisttestingtheseverityevaluationcircumventsbothfallaciesbutnosuchprincipled
remedyexistsfortheotherapproaches.
1. Introduction. The large n problem was initially raised by Lindley
ð1957ÞinthecontextofthesimpleNormalmodel,
X ∼NIIDðv;j2Þ; k51;2;:::;n;:::; ð1Þ
k
where NIIDðv;j2ÞstandsforNormal,Independent,andIdenticallyDistrib-
uted with mean v∈ð2`;`Þ and variance j2>0 ðassumed knownÞ, by
pointing out
ðaÞThelargenproblem.Frequentisttestingissusceptibletothefallacious
resultthatthereisalwaysalargeenoughsamplesizenforwhichanysim-
pleðpointÞnullhypothesis,sayH :v5v ,willberejectedbyafrequentist
0 0
a-significanceleveltest.
ReceivedApril2012;revisedAugust2012.
*Tocontacttheauthor,pleasewriteto:DepartmentofEconomics,VirginiaTech,Blacksburg,
VA24061;e-mail:[email protected].
yThanksareduetoDeborahMayofornumerousdiscussionsonthesetopicsandtotwoanon-
ymousrefereesformanyusefulcommentsandsuggestions.
PhilosophyofScience,80(January2013)pp.73–93.0031-8248/2013/8001-0004$10.00
Copyright2013bythePhilosophyofScienceAssociation.Allrightsreserved.
73
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74 ARISSPANOS
Lindley went on to claim that this result is paradoxical because, when
viewedfromtheBayesianperspective,onecanshow
ðbÞtheJeffreys-Lindleyparadox.Forcertainchoicesof theprior,the
posteriorprobabilityof H ;givenafrequentista-significancelevelrejec-
0
tion,willapproachoneasn→`.
This result was later called the Jeffreys-Lindley paradox because the
broaderissueofconflictingevidencebetweenthefrequentistandBayesian
approacheswasfirstraisedbyJeffreysð1939/1961,359–60Þ.Claimsaandb
contrastthebehaviorofafrequentisttestðp-valueÞandtheposteriorprob-
abilityof H asn→`,whichbringsupapotentialforconflictbetweenthe
0
frequentistandBayesianaccountsofevidence:
ðcÞBayesiancharge1.“TheJeffreys-Lindleyparadoxshowsthatforin-
ferenceaboutv;p-valuesandBayesfactorsmayprovidecontradictoryev-
idence and hence can lead to opposite decisions” ðGhosh, Delampady,
and Samanta 2006, 177Þ.
ThispotentialconflictisgivenamoredistinctBayesianslantby
ðdÞBayesiancharge2.AhypothesisthatiswellsupportedbytheBayes
factorcanbeðmisleadinglyÞrejectedbyafrequentisttestwhennislarge
ðseeBergerandSellke1987,112–13;Howson2002,45–49Þ.
Theproblemofconflictingevidencepertainstothebroaderphilosophical
issueofgroundingstatisticalpracticeonsoundprinciplesofinferenceand
evidence.Whathasnotbeenadequatelyexplainedinthisliteratureiswhy,
giventherejectionofH byafrequentisttest,itsposteriorprobabilitygoing
0
tooneasn→`ðirrespectiveofthetruthorfalsityof H Þisconducivetoa
0
moresoundaccountofevidence.
Theprimaryobjectiveofthisarticleistoconsiderthisissuebycompar-
ing the frequentist, Bayesian, and likelihoodist accounts of inference and
evidence.Thediscussioncanbeseenaspartofawiderendeavortousethe
errorstatisticalperspectiveðMayo1996Þinanattempttohaveacloserlook
atseveralBayesianallegationsthathaveunderminedthecredibilityoffre-
quentiststatisticsinphilosophicalcirclesoverthepasthalfcentury.
Thebriefcommentsinsection2provideapreludetothediscussionthat
follows by clarifying certain key issues at the outset. Section 3 introduces
the large n problem in frequentist testing using a numerical example dis-
cussed by Stone ð1997Þ. This example is based on a very large sample
ðn5527,135Þthatisusedtobringoutthefallaciousclaimsassociatedwith
the Jeffreys-Lindley paradox without the technical difficulties of invoking
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AFRAIDOFTHEJEFFREYS-LINDLEYPARADOX? 75
limitingarguments asn→`. In sections 4 and 5, the Bayesian and likeli-
hoodistapproachesareappliedtothisexamplewithaviewtodemonstrate
thatbothapproachesarefarfromimmunetofallaciousresults,contraryto
thecurrentview amongproponentsoftheBayesian andlikelihoodistper-
spectivesðseeBerger1985;Royall1997;Robert2007;interaliaÞ.Section6
illustrates how the postdata severityevaluationofthe p-valueandaccept/
rejectresultsaddressesnotonlythelargenproblembutalsothebroaderfal-
laciesofacceptance/rejectionandcallsintoquestionthechargesstemming
fromtheJeffreys-Lindleyparadox.Theseverityperspectiveisthenusedin
section7toshedlightonwhytheBayesianandlikelihoodistaccountsofev-
idencegiverisetohighlyfallaciousresults.
2. ClarifyingWhatIsFallaciousorParadoxical. Beforediscussingthe
variousclaimsthatrelatetotheJeffreys-Lindleyparadox,itisimportantto
bring out certain key issues that have not been adequately illuminated by
theliterature.
First,infrequentisttesting,whichincludesbothFisher’ssignificanceand
theNeyman-Pearsontesting,thelargenproblemarisesnaturallybecausethe
powerofany“good”ðconsistentÞtestincreaseswith n.Ana-significance
levelNeyman-Pearsontestissaidtobeconsistentwhenitspowertodetect
any discrepancy g≠0 from H approaches one as n→`: In this sense,
0
there is nothing fallacious or paradoxical about a small p-value or a rejec-
tion of the null, for a given significance level a;when n is large enough,
since a highly sensitive test is likely to pick up on tiny ðin a substantive
senseÞ discrepancies from H .
0
Second,Bayesiancharge2ðdÞoverlooksthefactthatthecornerstoneof
Neyman-Pearsontestingisthetrade-off betweenthetypeIðrejectH when
0
trueÞand type II ðaccept H when falseÞerror probabilities. In this sense,
0
these charges ignore the decrease in the type II error probability as n in-
creases,since,foragivendiscrepancyg,thepowerisoneminusthetypeII
errorprobability.Indeed,variousattemptshavebeenmadetoalleviatethe
largenproblem,includingdecreasingaasnincreasesinanattempttocoun-
terbalancetheincreaseinpowerassociatedwithnðseeLehmann1986Þ.The
difficulty,however,isthatonlycruderulesofthumbforadjustingacanbe
devised because the power of a test depends on other factors besides n.
Third,thelargenproblemðaÞconstitutesanexampleofabroaderprob-
lemknownasthefallacyofrejection:ðmisÞinterpretingrejectH ðevidence
0
againstH ÞasevidenceforaparticularH ;thiscaneasilyarisewhenatest
0 1
hasveryhighpower.Duetothetrade-off betweentypeIandIIerrorproba-
bilities,anyattempttoamelioratetheproblembyselectingasmallersignif-
icance level when n is large might render the result susceptible to the re-
versefallacyknownasthefallacyofacceptance:ðmisÞinterpretingacceptH
0
ðnoevidenceagainstH ÞasevidenceforH ;thiscaneasilyarisewhenatest
0 0
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76 ARISSPANOS
hasverylowpowerðe.g.,nisverysmallÞ.Asarguedbelow,thelargenprob-
lem ðaÞ, when a rejection of H isinterpretedasevidenceforH ,andthe
0 1
Jeffreys-LindleyparadoxðbÞconstituteexamplesof thefallaciesof rejec-
tionandacceptance,respectively.
The main argument of this article can be stated succinctly as follows:
ðiÞAlthoughthereisnothingfallaciousaboutasmallp-value,orarejection
of H ;whennislargeðitisafeatureofagoodfrequentisttestÞ,ðiiÞthereisa
0
problemwhensuchresults aredetachedfromthe testitself andaretreated
asprovidingthesameevidenceforaparticularalternativeH regardlessof
1
thegenericcapacityðthepowerÞofthetestinquestion.Thelargenproblem
isdirectlyrelatedtopointsðiÞandðiiÞbecausethepowerdependscruciallyon
n:Thatinturnrendersarejectionof H withalargenðhighpowerÞverydif-
0
ferentinevidentialtermsfromarejectionof H withasmallnðlowpowerÞ.
0
Thatis, therealproblemdoes notlie withthe p-valueortheaccept/reject
rules as suchbutwithhow such resultsare fashioned into evidence foror
against a particular hypothesis or an inferential claim relating to H ðH Þ.
0 1
ðiiiÞItisarguedthatthelargenproblemcanbecircumventedbyusingthe
postdataseverityassessmenttoprovideasoundevidentialaccountforfre-
quentistinference.Whetherdatax provideevidencefororagainstapartic-
0
ular hypothesisðH orH Þdepends cruciallyonthecapacityofthetestin
0 1
questiontodetectdiscrepanciesfromthenull.Thisstemsfromtheintuition
thatasmallp-valueorarejectionof H basedonatestwithlowpowerðe.g.,
0
a small nÞ for detecting a particular discrepancy g provides stronger evi-
denceforthepresenceofgthanusingatestwithmuchhigherpowerðe.g.,
alargenÞ.AsfirstpointedoutbyMayoð1996Þ,thisintuitioniscompletelyat
oddswith the Bayesian and likelihoodist intuitions articulated by Berger
andWolpertð1988Þ,HowsonandUrbachð2006Þ,andSoberð2008Þ.Indeed,
Mayowentontoproposeafrequentistevidentialaccountbasedonharness-
ingthisperceptiveintuitionintheformofapostdataseverityevaluationof
theaccept/rejectresults.Thisisbasedoncustomtailoringthegenericcapac-
ityofthetesttoestablishthediscrepancygwarrantedbydatax .Thiseviden-
0
tialaccountcanbeusedtocircumventtheabovefallacies,aswellasother
ðmisplacedÞ charges against frequentist testing ðsee Spanos 2011Þ. ðivÞ In
contrasttofrequentisttesting,nosuchreasonedremedyexistsfortheBayes-
ian andlikelihoodist approaches,whoseevidential accounts are shown to
beequallyvulnerabletothesefallacies.Astrongcasecanbemade,orsoit
is argued,thatthefallaciousnatureofthe Jeffreys-Lindleyparadoxstems
primarilyfromthefactthattheBayesianandlikelihoodistapproachesdis-
misstherelevanceofthegenericcapacityoftheparticulartestintheirevi-
dentialaccounts.Indeed,theseaccountscastasidetherelevanceof thesam-
ple space beyond x and “controlling” the relevant error probabilities for
0
beingthekeyweaknessesofthefrequentistapproach.
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AFRAIDOFTHEJEFFREYS-LINDLEYPARADOX? 77
3. The Large n Problem in Frequentist Testing. The approach to fre-
quentiststatisticsfollowedinthisarticleisknownaserrorstatisticsðMayo
1996Þ.It can be viewed as a refinement/extension of the Fisher-Neyman-
Pearson approach that offers a unifying inductive reasoning for frequentist
inference. It refines the Fisher-Neyman-Pearson approach by bringing out
the importanceofspecifyingexplicitlyaswellasvalidatingvis-à-visdatax
0
theinductivepremisesofinference.ItextendstheFisher-Neyman-Pearson
approachby supplementing it with a postdata severity assessment with a
viewtoaddressanumberoffoundationalproblemsrelatingtoasoundev-
idential account ðsee Mayo and Spanos 2004, 2006, 2011Þ.
Considerthe followingnumerical example discussedbyStoneð1997Þ:
Aparticle-physicscomplexplanstorecordtheoutcomesofalargenumber
ofindependentparticlecollisionsofaparticulartype,wheretheoutcomes
areeithertypeAortypeB....Theresultsaretobeusedtotestatheoreti-
calpredictionthattheproportionoftypeAoutcomes,h,isprecisely1/5,
againstthevaguealternativethathcouldtakeanyothervalue.Theresults
arrive:106,298typeAcollisionsoutof527,135.ð263Þ
How can one test this substantive hypothesis of interest?
Thefirststepistoembedtheabovematerialexperimentintoastatistical
model and frame the substantive hypothesis in terms of statistical param-
eters. With that in mind, let us assume that each of these n trials can be
viewed as a realization of a sample ðX ; X ;:::; X Þ of IID random vari-
1 2 n
ables defined by
(
1 if typeAcollisionoccurs;
X 5
0 if typeBcollisionoccurs;
whichtransformstheobservedsequenceofparticlesintodatax :5ð1,0,0,
0
1, 1, . . . , 0Þ. These conditions render the simple Bernoulli ðBerÞmodel,
X ∼BerIID½v;vð12vÞ(cid:2); v∈½0;1(cid:2); k51;2;:::;n;:::; ð2Þ
k
appropriateasastatisticalmodelinthecontextofwhichthematerialexper-
iment canbe embedded.Note that inpracticeone should validate the IID
assumptionstosecurethereliabilityofinference.Thesubstantivehypothesis
ofinterest,h5.2,canbeframedintermsoftheNeyman-Pearsonstatistical
hypotheses:
H∶v5v versus H∶v≠v ; for v 5:2; ð3Þ
0 0 1 0 0
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78 ARISSPANOS
specifiedsolelyintermsoftheunknownparameterof thestatisticalmodel
in ð2Þ.Aproperframingof theNeyman-Pearsonhypothesesrequiresapar-
titioningoftheparameterspace,irrespectiveofwhetheroneissubstantively
interestedinoneormorespecificvaluesof v.Fromthestatisticalperspec-
tive,allvaluesof vin½0;1(cid:2)arerelevantfordefiningtheoptimalityof thetest
ðseeSpanos2010,569Þ.
ThetestT :5fdðXÞ;C ðaÞg,definedby
a 1
pffiffiffi
nðX 2v Þ
dðXÞ5 pffiffiffiffiffiffiffinffiffiffiffiffiffiffiffiffiffi0ffiffiffi ∼H0 Binð0;1;nÞ;
v0ð12v0Þ ð4Þ
C ðaÞ5fx∶jdðxÞj ≥ c g;
1 ða=2Þ
whereX 5ð1=nÞon X and C ðaÞ; c denote the rejection region and
n i51 i 1 ða=2Þ
value,respectively,isauniformlymostpowerfulunbiasedNeyman-Pearson
testðseeLehmann1986Þ.Usingð4ÞonecandefinethetypeIerrorprobabil-
ityðsignificancelevelÞ:
PðjdðXÞj>c ;H Þ5a: ð5Þ
ða=2Þ 0
Thesamplingdistributioninð4ÞisbasedonthefactthatforaBernoulliIID
sampleX:5ðX,X,...,XÞ,therandomvariable Y 5nX 5on X;where
1 2 n n i51 i
Y denotes the number of 1’s in n trials, is binomially ðBinÞ distributed:
(cid:3) (cid:4)
fðy;v;nÞ5 n vyð12vÞn2y; y50;1;2;:::;n:
y
In light of the large sample size ðn 5 527,135Þ, it is often judicious to
chooseasmalltypeIerrorðLehmann1986Þ,saya5.003,whichyieldsa
rejectionvalueofc 52:968;notethattheNormalapproximationtothe
ða=2Þ
binomialdistributionisquiteaccurateinthiscase.
ThepoweroftestT atv 5v 1g definedby
a 1 0 1
Pðv Þ5PðjdðXÞj>c ; v 5v 1g Þ;
1 ða=2Þ 1 0 1
forv ∈V ;
1 1
as well as the type II error probability bðv Þ512Pðv Þ; for v ∈V , are
1 1 1 1
evaluatedusingthesamplingdistributionofdðXÞunderH ,whichtakesthe
1
form:
dðXÞv5∼v1 Binðdðpv1ffiÞffiffi;Vðv1Þ;nÞ; forv1∈V1:5V2f:2g;
nðv 2v Þ v ð12v Þ ð6Þ
wheredðv Þ5 pffiffiffiffiffiffi1ffiffiffiffiffiffiffiffiffiffi0ffiffiffiffi and Vðv Þ5 1 1 :
1 v0ð12v0Þ 1 v0ð12v0Þ
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AFRAIDOFTHEJEFFREYS-LINDLEYPARADOX? 79
This indicates that test Ta is consistent becausepitffisffiffi power increases with
dðv Þ—a monotonically increasing function of n—approaching one as
1
n→`ðseeLehmann1986Þ.Hence,otherthingsbeingequal,increasingn
increases the power of this test, confirming that there is nothing paradox-
ical aboutalarger nrenderingaðconsistentÞtestmoresensitive.
ApplyingtheNeyman-PearsontestT totheabovedata,
a
106;298
x 5 50:20165233;
n 527;135
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ
527;135½ð106;298=527;135Þ2:2(cid:2)
dðx Þ5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 52:999;
0 :2ð12:2Þ
leads to a rejection of H .The p-value, defined as the probability of ob-
0
serving an outcome x∈f0;1gn thataccordslesswellwithH thanx does
0 0
when H istrue,confirmstherejection of H :
0 0
PðjdðXÞj>jdðx Þj;H Þ5pðx Þ5:0027: ð8Þ
0 0 0
Thisdefinitionispreferredtothetraditionalone,“theprobabilityof observ-
ingaresultmoreextremethanx underH ,”becauseitblendsinbetterwith
0 0
thepostdataseverityevaluationdiscussedinsection6.Theresultpðx Þ5
0
:0027suggeststhatdatax indicate“some”discrepancybetweenv andthe
0 0
“true” v ðthat gave rise to x Þ,butitprovidesnoinformationaboutitsmag-
0
nitude.
As mentioned above, what is problematic is the move from the accept/
rejectresults,andthep-value,toclaimingthatdatax provideevidencefora
0
particular hypothesis, becausesuch a move is highlyvulnerable to the fal-
lacies of acceptance and rejection. However, in the context offrequentist
testing,thisvulnerabilitycanbecircumventedusingapostdataseverityeval-
uation.
How does the Jeffreys-Lindley paradox arise in this context? Using a
spikedpriordistributionðLindley1957Þoftheform
pðv5v Þ5p and pðv≠v Þ512p ; ð9Þ
0 0 0 0
theformalclaimassociatedwiththisparadoxis
(cid:3) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(cid:4)
p H jX 5v 1c v ð12v Þ=n →1 as n→ `;
0 n 0 ða=2Þ 0 0
thatis,theposteriorprobabilityof H ;conditionalondðx Þ5c ,goesto
0 0 ða=2Þ
one as n approaches infinity. What is not so obvious is why this result is
consideredanindicatorofthesoundnessoftheBayesianaccountofevi-
dence.
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80 ARISSPANOS
In relationto the natureofthe nullhypothesisðsimpleorcompositeÞ, it
is important to elucidate certain issues raised in this literature. First, in the
case of two composite hypotheses
H∶v≤v versus H∶v>v ; where v 5:2; ð10Þ
0 0 1 0 0
the test T*:5fdðXÞ;C*ðaÞg;whereC*ðaÞ5fx:dðxÞ≥c g;isbothconsis-
a 1 1 a
tent and uniformly most powerful ðsee Lehmann 1986Þ. In this sense, the
largenproblemhasnothingtodowiththerestrictiontoapointnullhypoth-
esis;it is simply a feature ofa consistent Neyman-Pearson test.Thenature
ofthenull ðsimpleorcompositeÞ only matters for thebehaviorof the like-
lihoodist ratio, the Bayes factor, and the posterior probability of H . Sec-
0
ond,thefactthatthediscrepancybetweenthep-valueandtheBayesfactor
is smaller for a composite null is not particularly interesting because both
measuresaremisleadingindifferentways.
Third, the comparison of the posterior pðvjx Þ;asvvariesover½0;1(cid:2)
0
ðwhichrepresentone’srevisedbeliefsaboutvinlightofx Þ,witherrorprob-
0
abilitiesðwhichmeasurehowoftenafrequentist procedureerrsasxvaries
overf0;1gnÞ,isdubious.Indeed,asarguedbyCasellaandBergerð1987,110Þ,
the comparison is made possible by using a blatant misinterpretation of the
p-value:“Thephrase‘theprobabilitythatH istrue’hasnomeaningwithin
0
frequency theory, but it has been argued that practitioners sometimes at-
tachsuchameaningtothe pvalue.Sincethepvalue,in thecasesconsid-
ered,isanupperboundontheinfimum ofPrðH jxÞit lies within or at the
0
boundary of a range of Bayesian measures of evidence demonstrating the
extenttowhichtheBayesianterminologycanbeattached.”
4. TheBayesianApproach. ConsiderapplyingtheBayesfactorprocedure
tothehypothesesð3Þusingauniformprior:
v∼Uð0;1Þ; thatis;pðvÞ51 for allv∈½0;1(cid:2): ð11Þ
ThisgivesrisetotheBayesfactor:
BFðx ;v Þ5 ELðv0;x0Þ
0 0 1
Lðv;x Þdv
0
0(cid:5) (cid:6)
527;135
ð:2Þ106;298ð12:2Þ527;1352106;298
5 E (cid:5)1(cid:5)06;298 (cid:6) (cid:6) ð12Þ
1 527;135
v106;298ð12vÞ527;1352106;298 dv
106;298
0
:000015394
5 58:115:
:000001897
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AFRAIDOFTHEJEFFREYS-LINDLEYPARADOX? 81
ItisinterestingtonotethatthesameBayesfactorð12Þarisesinthecaseof
thespikedpriorð9Þwith p 5:5;wherev5v isgivenpriorprobabilityof
0 0
:5andtheotherhalfisdistributedequallyamongtheremainingvaluesof v.
This is because for p 5:5,theratioðp =ð12p ÞÞ51andwillcancelout
0 0 0
fromBFðx ;vÞ.
0
ThenextstepinBayesianinferenceistouseð12Þasthebasisforfashion-
inganevidentialaccount.ABayesfactorresultBFðx ;v Þ>k;for k≥3:2;
0 0
indicates that data x favorthenullwiththe“strengthofevidence”increas-
0
ing with k. In particular, for 3:2 ≤ k <10,theevidenceissubstantial,for
10 ≤ k<100theevidenceisstrong,andfork≥100isdecisiveðseeRobert
2007Þ.
Comparingtheresultinð12Þwiththep-valueinð8Þ,Stoneð1997,263Þ
pointed out: “The theoretician is pleased when the ½likelihood–Bayes-
minded(cid:2) statistician reports a Bayesfactor of 8 to1 in favour of his brain-
child,butthepleasureisalloyedwhenheuseshisown P-valuecookbook
to reveal the 3.00 standard deviation excess of type A outcomes.”
Acloserscrutinyof theevidentialinterpretationof theresultinð12Þsug-
geststhatitisnotasclear-cutasitappears.Thisisbecause,onthebasisof
thesamedatax ,theBayesfactorBFðx ;v Þ“favors”notonlyv 5:2but
0 0 0 0
eachindividualvaluev insideacertainintervalaroundv 5:2:
1 0
V :5½:199648; :203662(cid:2)⊂V :5V2f:2g; ð13Þ
BF 1
wherethesquarebracket indicatesinclusionoftheendpoint,in thesense
that,foreachv ∈V ;BFðx ;v Þ>1,thatis,
1 BF 0 1
E
1
Lðv ;x Þ> Lðv;x Þdv; for allv ∈V : ð14Þ
1 0 0 1 BF
0
Worse,certainvaluesvzinV arefavoredbyBFðx ;vzÞmorestronglythan
BF 0
v 5:2:
0
vz∈V :5ð:2;:20331(cid:2)⊂V ; ð15Þ
LR BF
where the left parenthesis indicates exclusion of the end point. It is im-
portant toemphasizethatthesubsetsV ⊂V ⊂Vexistforeverydatax ;
LR BF 0
andonecan locate them by trial and error. However, there is a much more
efficient way to do that. As shown in section 5, V canbedefinedasa
subset of V around the maximum likelihood estimaLtRe ðMLEÞbv ðx Þ5
MLE 0
:20165233:Thisisnotcoincidentalbecause,asMayoð1996,200Þpointed
out,v♦5 ^v ðx Þisalwaysthemaximallylikelyalternative,irrespectiveof
MLE 0
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82 ARISSPANOS
thenullorothersubstantivevaluesofinterest.Inthisexample,theBayesfactor
forH∶v5v♦ versusH∶v ≠ v♦ yields
0 1
(cid:5) (cid:6)
527;135
ð:20165233Þ106;298ð12:20165233Þ527;1352106;298
106;298
BFðx ;v♦Þ5 E (cid:5)(cid:5) (cid:6) (cid:6)
0 1
527;135
v106;298ð12vÞ527;1352106;298 dv
106;298
0
:0013694656
5 5721:911; ð16Þ
:000001897
indicating not only decisive evidence for v5v♦ butalsothat
v♦ isfavoredbyBFðx ;v♦Þmorethan89
0
721:911
≃ timesstronger thanv 5:2!
8:115 0
This result is an instance of the fallacy of acceptance in the sense that the
Bayes factor BFðx ;v Þ>8is misinterpreted as providing evidenceforH:
0 0 0
v 5:2 against any value of v in V :5V2f:2g; when in fact BFðx ;vzÞ
0 1 0
providesstrongerevidenceforcertainvaluesofvz inV ⊂V .
LR 1
Whatisthesourceoftheseconflictingevidence?Stoneð1997,263Þcon-
jecturedthefollowingexplanation:“Asubhypothesisstronglyrejectedbya
significancetestmaybestronglysupportedinposteriorprobabilityifthe
prior puts insufficient weight on the hypotheses of non-negligible likeli-
hood.”
Letusfleshoutthisconjecture.First,chooseV ⊂V asarangeofvalues
BF 1
of v thatBFðx ;vyÞfavorsinthesensegiveninð14Þ.Tothisrangeof values
0
the prior attributes insufficient weight since
E
:203662
dv5:004:
:199648
Second,onecanrelateV tobothanequal-tailBayesianð12aÞ5:9997
BF
credibleintervalaswellasthefrequentistð12aÞconfidenceinterval:
½^v ðXÞ2ð3:6267ÞSDð^v ðXÞÞ; ^vðXÞ 1ð3:6267ÞSDð^v ðXÞÞ(cid:2);
MLE MLE MLE MLE
whereSDð^v ðXÞÞdenotesthestandarddeviationof ^v ðXÞ:Inlightof
MLE MLE
this,onecanconsiderV to represent a range of values of v with nonneg-
BF
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