Fino PhD Lectures 2018 –Genoa, 9 march 2018

Inductive arguments, inductive fallacies and related biases. Part II: probability and causality

MargheritaBenziUniversitàdelPiemonteOrientale

TopicsWewilldiscussthreeproblems

•  Probabilisticfallacies

•  Thegambler’sfallacy•  Theconjunctionfallacy

•  Causalfallacies•  Simpson’sparadox

Butbeforestarting…

HasanyonefoundabouttheBaaderMeinhofeffect? Whatistheassociatedheuristic? Otherbiases?

•  I.Thegambler’sfallacy

•  Tointroducethesubject,let’slookavideo.Itisafragmentofthefilm“RosenkrantzandGuildesternaredead”(1990);itisdirectedbyTomStoppard,whowrotethethetragicomedy(stagedforthefirsttimein1966).TheactorsareTimRothandGaryOldman.

•  •  • https://www.youtube.com/watch?v=KchhSIVwMdY• 

Questions

• WhyisGuildensternworrying?(or:isthereanythingstrangeandabitdisturbing?)

• WhydoesGuildensternlendtwocoinstoRosenkrantz?• GUIL:“Aweakermanmightbemovedtore-examinehisfaith,ifinnothingelseatleastinthelawofprobability”

• Buthequotestwo“lawsofprobabilities”thatpointtooppositedirections.Whatlawsarethey?

“Law”I(The“lawofaverages”)•  GUIL(understanding):Games.(Flipsacoin.)Thelawofaverages,ifIhavegotthisright,meansthatifsixmonkeyswerethrownupintheairforlongenoughtheywouldlandontheirtailsaboutasoftenastheywouldlandontheir–

•  (Comment:hilariousconfusionof3different‘chanceset-up’s:coins,diceandmonkeysinthe“Infinitemonkeystheorem”)

‘Law’2(equalchance)•  GUIL:Timehasstoppeddead,andthesingleexperienceofonecoinbeingspunoncehasbeenrepeated156times...(Heflipsacoin,looksatit,tossesittoROS.)Onthewhole,doubtful.Or:aspectacularvindicationoftheprinciplethateachindividualcoinspunindividually(hespinsone)isaslikelytocomedownheadsastailsandthereforeshouldcausenosurprisethateachindividualtimeitdoes

•  Whythereisatensionbetweenthe“lawofaverages”andthislaw?

•  Whichoneiswrong?

• Answer:thes.c.“lawofaverages”isafallacy:thegambler’sfallacy,aka“Thegambler’sruin”,or“Montecarlo’sfallacy”,or“Thefallacyofthematurityofchances”:

•  Thisfallacyistheerroneousbeliefthatifapossibleoutcomeofarandomsequencedoesnotoccurforalongtime,thenitismorelikelythatitwilloccurinthenexttrial,oratleastonceinthenextfewtrials.

• Whyisitafallacy?

• Kolmogorov’sAxioms•  •  LetAbeanevent(orasentencedescribinganevent)-•  •  ForanyA,0≤Pr(A)≤1•  IfAiscertain,Pr(A)=1•  IfAandBareincompatible,Pr(AorB)=Pr(A)+Pr(B)

•  ConditionalprobabilityPr(A|B)“TheprobabilityofAgivenB”

•  ProbabilisticIndependence

AisprobabilisticallyindependentofBifProb(A|B)=Prob(A)•  TheprobabilityofA-assumedthatB-isnotdifferentfromtheprobabilityofAingeneral.

•  Inotherwords,BdoesnotaltertheprobabilityofA.

Ifweconsideragenuinechancesetup,i.e.somethingthatproducesgenuinerandomevents,liketossinganunbiasedcoin,oneoutcomedoesnotinfluencethenextone.Therefore,theyareprobabilisticallyindependent(bycontrast,knowingthatachildissixyearsoldinfluencesthelikelihoodthatshecanread;therefore,theageofthechildandherabilitytoreadarenotprobabilisticallyindependent)

•  Ifacoin,oraroulette,oradeckofcartsareunbiased,oneoutcomedoesnotinfluencesthenextone.Letusaconsiderthetossingofacoin.

•  H=head,T=tail.•  Ifthecoinisfair

Pr(T)=Pr(H)=)½ Pr(T|H)=Pr(T)=½

•  Wehavealso Pr(H|H)=½,•  And

Pr(H|HHHHHHHHHHHHHH)=Pr(T|HHHHHHHHHHHHHH)=½.•  Inotherterms,theprobabilityofHHHHHHHHHHHHHHHisthesameastheprobabilityof

HHHHHHHHHHHHHHT

•  Ifweknowforcertainthatthecoinisfair,thefactofhavingobtainedalongsequenceofheadsshouldnotlowerourdegreeofbeliefthatthenexttosswillhavetheoutcome‘head’.

• Butevenifweknowthisfact(“thecoin–ortheroulettewheel–“hasnomemory”:inthemovie,alsoRosenkrantzhasnomemory)weoftenfeelstronglyinclinedtothinkthat,thelongerthepastsequenceofheadshasbeen,themoreprobableistheoutcome‘tailofthenexttoss’.

• Perhapsthemostfamousexampleofthegambler’sfallacyoccurredinagameofrouletteattheMonteCarloCasinoonAugust18,1913,whentheballfellinblack26timesinarow.Thiswasanextremelyuncommonoccurrence,withaprobabilityofaround1in136.8million.Gamblerslostmillionsoffrancsbettingagainstblack,reasoningincorrectlythatthestreakwascausinganimbalanceintherandomnessofthewheel,andthatithadtobefollowedbyalongstreakofred.[Wikipedia]

• ButwhywethinkthatPr(HHHHHHHHHHHHHHT)>Pr(HHHHHHHHHHHHHHH)?Aplausibleansweristhatthefirstsequenceismoresimilartoournotionof‘randomsequence’;ifthisexplanationiscorrect,wehave,again,aninstanceoftheavailabilityheuristics.

II.Theconjunctionfallacy

•  Tversky,A.,&Kahneman,D.(1982).Judgmentsofandbyrepresentativeness.InD.Kahneman,P.Slovic,&A.Tversky(Eds.),Judgmentunderuncertainty:Heuristicsandbiases(pp.84–98).NewYork,NY:CambridgeUniversityPress.Tversky,A.,&Kahneman,D.(1983).Extensionalvs.intuitivereasoning:Theconjunctionfallacyinprobabilityjudgment.PsychologicalReview,90,293–3l5.

•  Lindais31yearsold,single,outspokenandverybright.Shemajoredinphilosophy.Asastudent,shewasdeeplyconcernedwithissuesofdiscriminationandsocialjustice,andalsoparticipatedinanti-nucleardemonstrationswhenshewasununiversitystudent

•  Choosethemostprobablehypotheses:

(a)Lindaisabankteller

(b)Lindaisabanktellerandisactiveinthefeministmovement

•  TverskyandKahnemandiscoveredthatpeoplewerestronglyorientedtojudge(b)moreprobablethan(a)(inparticular,inasampleof142studentswhowereaskedtoassessthecomparativeprobabilityofb)vsa),85%judgedb)moreprobablethana).

Thisanswerisfalse,accordingtoprobabilitycalculus.

• ConjunctionruleforanypairofeventsAandBtheprobabilityoftheirconjunctioncanneverbehigherthantheprobabilityofanyofthemalone.

Pr(A∧B)≤Pr(A),Pr(B).

•  AccordingtoProb.Calc.theprobabilityoftheconjunctionofAandBcanneverbegreaterthantheprobabilityofanyofitsconjuncts

•  examplestheprobabilitythatImetaFrenchmantodaycannotbesmallerthantheprobabilitythatImeetaFrenchmanwearingsunglasses;

•  theprobabilitythatgoingoutfromthisbuildingwemeetaladywithadogcannotbelowerthantheprobabilitythatwemeetaladywithadogthatisalabrador).

• Whydopeopleanswerthat(b)(FT)ismoreprobablethan(a)(T)?• Whatheuristicscouldbethecauseoftheconjunctionfallacy,andwhy?•  Doyouthinkthatthemorefrequentanswerisirrational,orcompletelywrong?•  Ifnot,why?

• Violationsofrationality• Psychology,philosophy,economy• Morewidespreadhypothesis:AvailabilityHeuristicsstrikesagain• Controversy:«Lindaindustry»

Gigerenzer

Thelawisnotviolated,however,ifparticipantsinthesestudiesunderstandtheword“probability”inasensedifferentfromtheoneassignedtoitbymodernprobabilitytheory.ThereissimilarlynoviolationifBisinterpretedtomeanT∧¬F,orisinterpretedinanywayotherthanasaconjunctofT∧F.

•  “AccordingtoHertwigandGigerenzer(1999,p.278),subjectsareevenurgedtochooseanon-mathematicalinterpretationbecauseofPaulGrice’srelevancemaxim.Thismaximisaconversationalruleusuallytakentobefollowedbyparticipantsinadialogue.Itsays,roughly,thatone’scontributionshavetoberelevanttothetopicandgoaloftheconversation.Appliedtothecaseathand,thismeansthateverypartoftheexperimenters’instructionisrelevant.Onthemathematicalinterpretationof‘probable’,however,thedescriptionofLindaisirrelevanttothequestionsubjectsaresupposedtoanswer.Hence,peoplemightchooseareadingof‘probable’notcapturedbyprobabilitytheorybecause,otherwise,thepersonalitysketchhadtobeconsideredidle.”[MarkSiebel.”There’ssomethingaboutLinda”Siebel,Mark."There’ssomethingaboutLinda:Probability,coherenceandrationality."InFirstSalzburgworkshoponparadigmsofcognition,Salzburg.2002.]

•  Likelihoodvsprobability

•  Pr(E|H)vsPr(H|E)

•  ConfirmationvsProbability

(Crupi,tentorieRusso2013)

(Theportraitconfirms‘feminist’morethanbankteller)Thestorycontinues

• Averyshortvisualbreak• Otherstatisticalfallacies:falseassociation

False statistical associations http://www.tylervigen.com/spurious-correlations

http://www.tylervigen.com/spurious-correlations

http://www.tylervigen.com/spurious-correlations

http://www.tylervigen.com/spurious-correlations

• PartIICausalfallacies

Causalfallaciesareinductivefallaciesthatresultinwrongcausaljudgements.manysubspecies‘posthocergopropterhoc’‘cumhocergopropterhoc’‘noncausaprocausa’apopularfallacyaboutafallacy–thetermisoftenusedasageneralizationovertheprecedingtwo,butthefallacyof‘noncausaprocausa’wasnotbornasaninductivefallacy,butasalogicalone.

• ParadossodiSimpson

Nel campione generale la proporzione di guariti tra chi prende la medicina è più alta della percentuale di guariti tra chi non prende la medicina. La med. sembra efficace

Popolaz. Guariti NonGuariti

Totale %guariti

Medicina 20 20 40 50%

NoMedicina

16 24 40 40%

36 44 80

Tra le donne la proporzione di guarite tra chi prende la medicina è più bassa della percentuale di guarite tra chi non prende la medicina. La med. non sembra efficace

Femmine Guarite NonGuarite

Totale %guarite

Medicina 2 8 10 20%

NoMedicina

9 21 30 30%

11 29 40

Tra i maschi la proporzione di guariti tra chi prende la medicina è più bassa della percentuale di guariti tra chi non prende la medicina. La med. non sembra efficace

Maschi Guariti NonGuariti

Totale %guariti

Medicina 18 12 30 60%

NoMedicina

7 3 10 70%

25 15 40

Simpson’s paradox

Adrugisinactiveinthesub-populationsButlooksbeneficialwrtthewholepopulation

Why?

DifferentproportionsanddimentionsManrecovernomatterhow(genderasconfounder)

LagraduateschooldiBerkeleyfuaccusatadiattuareunapoliticadiammissionechediscriminavaledonne.Diconseguenza,siponevalaseguentequestione:‘IlfattodiessereunadonnaèrealmenteunacausadinonammissioneaBerkeley?’.L’accusaapparivafondatasullabasedeidatiprobabilistici:laprobabilitàdiessereaccettatieramoltopiùaltaperimaschidiquantononlofosseperlefemmine.Tuttavia,Bickel,HammeleO’Connell[...]esaminaronoidaticonmaggioreattenzioneescoprironochequestocessavadivalerequandoripartivano[leammissioni]perdipartimento.Nellamaggiorpartedegliottantacinquedipartimentilaprobabilitàcheunadonnafosseammessaeraesattamentelastessadiunuomo,edinalcunidipartimentieraaddiritturapiùalta.[Cartwright1983,p.37]

Other inductive causal fallacies •  “jumping”fromassociationsorfromtemporalsuccessionstocausalrelations.•  cumhocergopropterhoc(constantconjunction)•  everytimeMariocomestothepicnic,itrains;•  posthocergopropterhoc•  theroostercausesthesunrisebecauseitstartssingingjustbeforeit.•  inversecausation•  Scratchingisthecauseoftheitchingofmyhand•  Selectionbias:•  wetendtothinkthatsomeactivities,asbeinganOlympicswimmer,oraprofessionaldancer,haveaverystrongeffectonthephysicalconstitutionoftheswimmersorthedancers,withoutconsideringthatifonehasnotacertaintypeofphysicalconstitutionitisverydifficultthatheorshebecomesanOlympicswimmeroraprofessionaldancer.

• mentendedtoapplytothedepartmentsthatarethehardesttogetinto

• mentendedtoapplytodepartmentsthatwereeasiertogetinto

•  Sowomenwererejectedmorethanmen.