Let the Evidence Speak: Using Bayesian Thinking in Law, Medicine, Ecology and Other Areas

220 Pages·2018·7.75 MB·English

by  Alan Jessop

Alan Jessop
Let
the
Evidence
Speak
Using Bayesian Thinking
in Law, Medicine,
Ecology and Other Areas
Let the Evidence Speak
Alan Jessop
Let the Evidence Speak
Using Bayesian Thinking in Law, Medicine,
Ecology and Other Areas
AlanJessop
DurhamUniversityBusinessSchool
DurhamUniversity
Durham,UnitedKingdom
ISBN978-3-319-71391-5 ISBN978-3-319-71392-2 (eBook)
https://doi.org/10.1007/978-3-319-71392-2
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Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
PartI Likelihood
2 WhoseCar?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Bayes’Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 TrackRecord. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 GameShow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 MarginofError. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 FiveThoughtsAboutLikelihoods. . . . . . . . . . . . . . . . . . . . . . . . . . 53
PartII BaseRate
8 Diagnosis. . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . 69
9 Information. . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . . .. . 83
10 BeingCarefulAboutBaseRates. . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11 Independence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
12 Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
PartIII Application
13 WhoWroteThat?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14 Wood,TreesandWildlife. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
15 RadiocarbonDating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
v
vi Contents
16 BayesandtheLaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
17 TheSelectionTask. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
18 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
AFormulaforBayes’Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Chapter 1
Introduction
Bayesisalltherage!
You’llreadthis,orsomethinglikeit,fromtimetotime.Thisexampleisfromthe
openingsentence ofabookbyLucBovensandStephanHartmann[1].Infullthe
sentence reads “Bayes is all the rage in philosophy”. You may not be much
interested in philosophy, but that’s just one of the places where Bayes’ Rule
makesanappearance.Thereareothers,notonlyinbookswrittenforacademics.
For instance, Angela Saini wrote a piece in The Guardian about the use of
Bayes’RuleinacourtoflawundertheheadlineTheformulaforjustice.Shewrites
thatBayes’Rulewas
Invented by an 18th-century English mathematician, Thomas Bayes, this calculates the
oddsofoneeventhappeninggiventheoddsofotherrelatedevents[2].
And here’s Tim Harford writing in a piece called How to make good guesses that
whenwehavetwopiecesofinformation
Logically,oneshouldcombinethetwopiecesofinformation...Thereisamathematical
rulefordoingthisperfectly(it’scalledBayes’rule)[3].
Finally,describingtheapplicationofBigDatatoonlineadvertising,CathyO’Neil
writes
ThedatascientistsstartoffwithaBayesianapproach,whichinstatisticsisprettycloseto
plainvanilla.ThepointofBayesiananalysisistorankthevariableswiththemostimpact
onthedesiredoutcome.Searchadvertising,TV,billboards,andotherpromotionswould
eachbemeasuredasafunctionoftheireffectivenessperdollar.Eachdevelopsadifferent
probability,whichisexpressedasavalue,oraweight[4].
Allofwhichisfine,butjustwhatisBayes’Rule?
It’s been with us for about 300 years and every so often enjoys some popular
prominence.Articlesandbookssuchasthoseabovearenotwrittenforatechnical
audience. They may have whetted your appetite—sounds good but how does it
work? You want to know more but you don’t want a statistics text. This book is
foryou.
~ (cid:129) ~
©SpringerInternationalPublishingAG,partofSpringerNature2018 1
A.Jessop,LettheEvidenceSpeak,https://doi.org/10.1007/978-3-319-71392-2_1
2 1 Introduction
Decisionsshouldbemadebasedonevidence.
AdoctormayuseatestsuchasabloodtestorMRIscanwhendiagnosingwhat
illnessapatientmighthave.Experimentalresultsshowthediagnosticpowerofthe
test,itsabilitytoprovideevidenceforthedoctorwhothenmustdecidewhattotell
thepatient.Sometestsareveryinformative,somelessso.
Police investigating a crime and juries trying an accused have to use evidence
too. This is of variable quality, ranging from DNA matching (pretty good) to
eyewitness identification (not so good), and yet decisions of guilt and innocence
aremade.
Whatiscommonisthatsomeone—adoctor,ajuror—makesadecisionbasedon
what they believe to be true and this belief is based, in part at least, on evidence.
Bayes’Ruleprovidesthenecessarylinkbetweentheevidenceandwhattobelieve
based on that evidence. The better the evidence the stronger is the belief that the
patient is ill or the accused is guilty. The strength of the evidence, and so of our
belief,isdescribedusingprobability.
Bayes’Ruletakesusthatfar.Whattodo,whatactiontotake,isyourdecision.
Youwillhaveameasureofyourjustifiedbelieftohelpyou.
~ (cid:129) ~
TherearemanyfinebooksandresearchpapersyoucouldreadaboutBayes’Rule
and its applications but they tend to be written in the formal language of mathe-
matics.Theymayassumethatyouknowmorethanyoudo.Weallhitabarrierat
somepoint.
Themotivationforthisbookisthatitispossibleandusefultodescribeforany
reader,whatevertheir(whichistosay,your)background,whatBayes’Ruleisand
howitworks.Bayes’Ruleisusuallypresentedasaformula.Formanythisisnotan
encouragementtoreadfurther.Butagreatmanyrealapplications,notjusttextbook
illustrations,neednomorethansimplearithmetic.ToshowthecalculationsIhave
used a table—the Bayes Grid. All you need to know is how to add, multiply and
divide.Thinkofaverysimplespreadsheet.
Bayes’Ruleisalsoawayofthinking,offorcingyoutoanswerquestions
WhatdoIknowaboutthisproblem?
WhatevidencedoIhaveandhowgoodisit?
Whatalternativeexplanationsmightaccountfortheevidence?
Asking the right questions is always a good idea, even if you sometimes need
helpwiththemaths.
~ (cid:129) ~
References 3
This book is organised in three parts. The first sets out the basic relation between
evidence and the alternative accounts for that evidence. The second extends the
basicframeworktobringinotherinformationyoumayhave.Thismaybebasedon
dataoronjudgement.Youneedtoknowhowtodealwiththeissuesraisedbythis
usefulextension.
Up to this point the calculations have been easy. The emphasis has been on
thinkingabouthowtouseevidence,askingtherightquestions.Manyapplications
arenotsostraightforwardeitherbecausethemathsismoredifficultorbecausethe
decision problem is, by its nature, more complex. The chapters in this third part
coverboth.
Insummary,thethreepartsare
Likelihood The key to it all. The necessary description of the relation between
theevidenceyouhave,whatmightaccountforit,andwhatyouare
justifiedinbelieving.
Baserate Baseratesenableyoutousecontextualorjudgementalinformation
as well as evidence. Bayes’ Rule becomes a means of learning,
modifying your starting belief in light of new evidence. Bringing
injudgementisquitenatural,butcareisneeded.Wemaynotbeas
good as we think at expressing our judgement in the form of
probabilities.
Application Many of the applications of Bayes’ Rule use mathematical models
which are daunting for non-mathematicians. However, the
underlying principles are the same as in the earlier parts. These
final chapters show how some Bayes’ thinking helps structure
problemsandsomeofthemodellingissueswhichfollow.
The closing two chapters, on law and the psychology of
reasoning,donotinvolve heavy maths. Theyfurther emphasisethe
benefitsofusingBayes’Ruleasawayofthinking.
Notesandreferencesaretheretoprovidehintsandreadingshouldyouwishtogo
further.Someofthereferencesmaynotbeforyoubutnoneareneededforyouto
readthisbookand,Ihope,benefitfromit.
~~~ (cid:129) ~~~
References
1.BovensL,HartmannS(2003)Bayesianepistemology.Clarendon,Oxford,pv
2.SainiA(2011)Theformulaforjustice.GuardianMagazine,2Oct2011,p12
3.HarfordT(2016)Howtomakegoodguesses.FinancialTimesWeekendMagazine,27/28Feb,
p45
4.O’NeillC(2016)Weaponsofmathdestruction:howbigdataincreasesinequalityandthreatens
democracy.AllenLane,London,p74
Part I
Likelihood