Dating Hypatia's birth: a probabilistic model

25 Pages·2017·0.98 MB·English

NISSUNAUMANAINVESTIGAZIONE SI PUO DIMANDARE VERASCIENZIA
S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI
LEONARDO DAVINCI
vol. 5 no. 1 2017
Mathematics and Mechanics
of
Complex Systems
CANIO BENEDETTO, STEFANO ISOLA AND LUCIO RUSSO
DATING HYPATIA’S BIRTH: A PROBABILISTIC MODEL
msp
MATHEMATICSANDMECHANICSOFCOMPLEXSYSTEMS
Vol.5,No.1,2017
M ∩ M
dx.doi.org/10.2140/memocs.2017.5.19
DATING HYPATIA’S BIRTH: A PROBABILISTIC MODEL
CANIO BENEDETTO, STEFANO ISOLA AND LUCIO RUSSO
Weproposeaprobabilisticapproachasadatingmethodologyforeventslikethe
birthofahistoricalfigure. Themethodisthenappliedtothecontroversialbirth
dateoftheAlexandrianscientistHypatia,provingtobesurprisinglyeffective.
1. Introduction 19
2. Aprobabilisticmethodforcombiningtestimonies 20
3. ApplicationtoHypatia 25
4. Conclusions 38
References 39
1. Introduction
Although in historicalinvestigation itmay appear meaningless to doexperiments
onthebasisofapreexistingtheory—andinparticular,itdoesnotmakesenseto
provetheoremsofhistory—itcanmakeperfectsensetouseformsofreasoning
typicaloftheexactsciencesasanaidtoincreasethedegreeofreliabilityofapar-
ticularstatementregardingahistoricalevent. Thispaperdealswiththeproblemof
datingthebirthofahistoricalfigurewhentheonlyinformationavailableaboutitis
indirect—forexample,asetoftestimonies,orscatteredstatements,aboutvarious
aspectsofhis/herlife. Thestrategyisthenbasedontheconstructionofaprobability
distribution forthe birthdateout ofeach testimonyand subsequentlycombining
thedistributionsso obtainedinasensible way. Onemightraise several objections
to this program. According to Charles Sanders Peirce [1901], a probability “is
theknownratiooffrequencyofaspecificeventtoagenericevent”,butabirthis
neitheraspecificeventnoragenericeventbutan“individualevent”. Nevertheless,
probabilistic reasoningis used quite oftenin situations dealingwith events that can
beclassifiedas“individual”. Inprobabilisticforecasting,onetriestosummarize
whatisknownabout futureeventswiththeassignmentofaprobability toeachof
anumber ofdifferentoutcomesthat areoftenevents ofthiskind. Forinstance, in
sport betting, a summary of bettors’ opinions about the likely outcome of a race
CommunicatedbyFrancescodell’Isola.
MSC2010: 01A20,62C05,62P99.
Keywords: dating,probabilisticmethod,historicaltestimonies,decisiontheory,Hypatia.
19
20 CANIOBENEDETTO,STEFANOISOLAANDLUCIORUSSO
isproducedinordertosetbookmakers’pay-offrates. Bytheway,thistypeofob-
servationliesatthebasisofthetheoreticalformulationofthesubjectiveapproach
in probability theory [de Finetti 1931]. Although we do not endorse de Finetti’s
approachinallitsimplications,weembraceitsseverecriticismoftheexclusiveuse
ofthefrequentistinterpretationintheapplicationofprobabilitytheorytoconcrete
problems. In particular, we feel entitled to look at an “individual” event of the
historicalpastwithaspiritsimilartothatwithwhichonebetsonafutureoutcome
(this is a well known issue in the philosophy of probability; see, e.g., [Dubucs
1993]). Plainly, as the information about an event like the birth of an historical
figure is first extracted by material drawn from various literary sources and then
treatedwithmathematicaltools,bothourapproachandgoalareinterdisciplinary
intheiressence.
2. Aprobabilisticmethodforcombiningtestimonies
Let X = [x−,x+] ⊂ (cid:90) be the time interval that includes all possible birth dates
of a given subject (terminus ad quem). X can be regarded as a set of mutually
exclusive statements about a singular phenomenon (the birth of a given subject
in a given year), only one of which is true, and can be made a probability space
(X,F,P ), with F the σ-algebra made of the 2|X| events of interest and P the
0 0
uniformprobabilitymeasureonF(referencemeasure): P (A)=|A|/|X|(where
0
|A| denotes the number of elements of A). In the context of decision theory, the
assignmentofthisprobabilityspace canberegardedastheexpressionofabasic
stateofknowledge,intheabsenceofanyinformationthatcanbeusedtodiscrimi-
nateamongthepossiblestatementson thegivenphenomenon,namelyasituation
inwhichLaplace’sprincipleofindifferencecanbelegitimatelyapplied.
Now suppose we have k testimonies T , i = 1,...,k, which in first approx-
i
imation we may assume independent of each other, each providing some kind
of information about the life of the subject, and which can be translated into a
probability distribution p on F so that p (x) is the probability that the subject
i i
is born in the year x ∈ X based on the information given by the testimony T ,
i
assumedtrue,alongwithsupplementaryinformationsuchas,e.g.,lifetablesfor
thehistoricalperiodconsidered. Theprecisecriteriafortheconstructionofthese
probabilitydistributionsdependsonthekindofinformationcarriedbyeachtesti-
mony andwillbe discussedcase bycasein thenext section. Ofcourse, weshall
alsotake intoaccountthepossibilitythat sometestimoniesarefalse, therebynot
producing any additional information. Wemodel this possibility by assuming that
thecorrespondingdistributionsequalthereferencemeasure P .
0
Theproblemthatwewanttodiscussinthissectionisthefollowing: howcanone
combinethedistributions p insuchawaytogetasingleprobabilitydistribution Q
i
DATINGHYPATIA’SBIRTH:APROBABILISTICMODEL 21
thatsomehowoptimizes theavailable information? Toaddress thisquestion, letus
observe that from the k testimonies taken together, each one with the possibility
to be true or false, one gets N =2k combinations, corresponding to as many bi-
narywordsσ =σ (1)···σ (k)∈{0,1}k,whichcanbeorderedlexicographically
s s s
accordingtos =(cid:80)k σ (i)·2i−1∈{0,1,...,N −1},andgivenby
i=1 s
P (·)= (cid:81)ik=1 piσs(i)(·) , pσs(i)=(cid:26)pi, σs(i)=1, (2-1)
s (cid:80)x∈X (cid:81)ik=1 piσs(i)(x) i P0, σs(i)=0.
Inparticular,onereadilyverifiesthat P isbutthereferenceuniformmeasure.
0
Now,if(cid:127)denotestheclassofprobabilitydistributions Q : X →[0,1],welook
forapoolingoperator T :(cid:127)N →(cid:127)thatcombinesthedistributions P byweighing
s
them in a sensible way. The simplest candidate has the general form of a linear
combination
N−1 N−1
(cid:88) (cid:88)
T(P0,...,PN−1)= wsPs, ws ≥0, ws =1, (2-2)
s=0 s=0
which, as we shall see, can also be obtained by minimizing some information-
theoreticfunction.
Remark2.1. Theissuewearediscussingherehasbeentheobjectofavastamount
ofliteratureregardingthenormativeaspectsoftheformationofaggregateopinions
in several contexts (see, e.g., [Genest and Zidek 1986] and references therein).
In particular, it has been shown by McConway [1981] that, if one requires the
existenceofafunction F :[0,1]N →[0,1]suchthat
T(P0,...,PN−1)(A)= F(P0(A),...,PN−1(A)) forall A∈F (2-3)
with P (A)=(cid:80) P (x), then whenever |X|≥3, F must necessarily have the
s x∈A s
formofalinearcombinationasin(2-2). Theaboveconditionimpliesinparticular
that the value of the combined distribution on coordinates depends only on the
corresponding values on the coordinates of the distributions P , namely that the
s
poolingoperatorcommuteswithmarginalization.
However,somedrawbacksofthelinearpoolingoperatorhavealsobeenhigh-
lighted. Forexample,itdoesnot“preserveindependence”ingeneral: if|X|≥5,
itisnottruethat P (A∩B)= P (A)P (B),s =0,...,N −1,entails
s s s
T(P ,...,P )(A∩B)=T(P ,...,P )(A)T(P ,...,P )(B)
0 N−1 0 N−1 0 N−1
unless w =1 for some s and 0 for all others [Lehrer and Wagner 1983; Genest
s
andWagner1987].
(Anotherformofthepoolingoperatorconsideredintheliteraturetoovercome
thedifficultiesassociatedwiththeuseof(2-2)isthelog-linearcombination
22 CANIOBENEDETTO,STEFANOISOLAANDLUCIORUSSO
N−1 N−1
T(P0,...,PN−1)=C (cid:89) Psws, ws ≥0, (cid:88)ws =1, (2-4)
s=0 s=0
whereC isanormalizingconstant[GenestandZidek1986;Abbas2009].)
On the otherhand, inour context, the independence preservation propertydoes
not seem so desirable: the final distribution T(P0,...,PN−1) relies on a set of
information much wider than thatassociated with the single distributions P , and
s
one can easily imagine how the alleged independence between two events can
disappearastheinformationaboutthemincreases.
2.1. Optimization. The linear combination (2-2) can also be viewed as the mar-
ginal distribution1 of x ∈ X under the hypothesis that one of the distributions
P0,...,PN−1 isthe“true”one(withoutknowingwhich)[GenestandMcConway
1990]. Inthisperspective,(2-2)canbeobtainedbyminimizingtheexpectedloss
ofinformationduetotheneedtocompromise,namelyafunctionoftheform
N−1
(cid:88)
I(w,Q)= w D(P (cid:107) Q)≥0, (2-5)
s s
s=0
where
(cid:88) (cid:18)P(x)(cid:19)
D(P (cid:107) Q)= P(x)log (2-6)
Q(x)
x∈X
istheKullback–Leiblerdivergence[1951],representingtheinformationlossusing
the measure Q instead of P. Note that the concavity of the logarithm and the
Jenseninequalityyield
(cid:88) P(x) (cid:88) Q(x)
− P(x)log ≤log P(x) =0
Q(x) P(x)
x x
andtherefore
D(P (cid:107) Q)≥0 and D(P (cid:107) Q)=0 ⇐⇒ Q ≡ P. (2-7)
Wehavethefollowingresult.
Lemma2.2. Givenaprobabilityvectorw=(w0,w1,...,wN−1),
(cid:88)
argminI(w,Q)= Qw ≡ wsPs. (2-8)
Q∈(cid:127) s
Moreover,
(cid:18) (cid:19)
(cid:88) (cid:88)
I(w,Qw)= H wsPs − wsH(Ps), (2-9)
s s
where H(Q)=−(cid:80) Q(x)logQ(x)istheentropyof Q ∈(cid:127).
x∈X
1Inthesensethatamarginalprobabilitycanbeobtainedbyaveragingconditionalprobabilities.
DATINGHYPATIA’SBIRTH:APROBABILISTICMODEL 23
Proof. Equation (2-8) can beobtained using themethod of Lagrange multipliers.
Analternativeargumentmakesuseoftheeasilyderived“parallelogramrule”:
(cid:80)wsD(Ps (cid:107) Q)=(cid:80)wsD(Ps (cid:107) Qw)+D(Qw (cid:107) Q) forall Q ∈(cid:127). (2-10)
s s
From (2-7), we thus get I(w,Qw)≤ I(w,Q) for all Q ∈(cid:127). The uniqueness of
theminimumfollowsfromthe convexityof D(P (cid:107) Q)withrespectto Q. Finally,
checking(2-9)isasimpleexercise. (cid:3)
Remark2.3. Itisworthmentioningthat,ifwetook(cid:80) w D(Q (cid:107) P )(insteadof
s s s
(cid:80) w D(P (cid:107) Q))asthefunctiontobeminimized(stillvarying Q withw fixed),
s s s
theninsteadofthe “arithmeticmean”(2-2),the“optimal”distributionwould have
beenthe“geometricmean”(2-4)(seealso[Abbas2009]).
2.2. Allocatingtheweights. Wehaveseenthatforeachprobabilityvectorwinthe
N-dimensionalsimplex{ws ≥0:(cid:80)sN=−01ws =1}thedistribution Qw =(cid:80)swsPs
isthe“optimal”one. Wearenowleftwiththeproblemofdeterminingasensible
choiceforw. Thiscannotbeachievedbyusingthesamecriterion,inthatby(2-7)
infw I(w,Qw)=0andtheminimumisrealizedwheneverws =1forsomes and
0forallothers.
Asuitableexpressionfortheweightsw canbeobtainedbyobservingthatthe
s
term (cid:80) (cid:81)k pσs(i)(x) is proportional to the probability of the event (in the
x∈X i=1 i
productspace X[1,k])thatthebirthdatesofk differentsubjects,withthei-thbirth
datedistributedaccordingto pσs(i),coincide,andthus,itfurnishesameasureofthe
i
degreeofcompatibilityofthedistributions p involvedintheproductassociated
i
withthewordσ .
s
Itthusappearsnaturaltoconsidertheweights
(cid:80) (cid:81)k pσs(i)(x)
w = x∈X i=1 i , (2-11)
s (cid:80)N−1(cid:80) (cid:81)k pσs(i)(x)
s=0 x∈X i=1 i
which,onceinsertedin(2-2),yieldtheexpression
(cid:80)N−1(cid:81)k pσs(i)(·)
T(P0,...,PN−1)(·)= (cid:80) s=(cid:80)0N−1i=(cid:81)1k i pσs(i)(x). (2-12)
x∈X s=0 i=1 i
Remark2.4. Thereareatleastk+1strictlypositivecoefficientsw . Theycorre-
s
spondtothewordsσ(i) withσ(i)(i)=1forsomei ∈{1,...,k}andσ(i)(j)=0for
s s s
j(cid:54)=i,plusonetotheword0k,thatis,tothedistributions Ps(i)≡pi,i∈{0,1,...,k},
where p ≡ P .
0 0
2.3. Weightsaslikelihoods. Asomewhatcomplementaryargumenttojustifythe
choice (2-11) for the coefficients w can be formulated in the language of prob-
s
abilisticinference,showingthattheycanbeinterpretedas(normalized)average
24 CANIOBENEDETTO,STEFANOISOLAANDLUCIORUSSO
likelihoodsassociatedwiththevariouscombinationscorrespondingtothewordsσ .
s
Moreprecisely,witheachpairof“hypotheses”oftheform
(cid:26){T true}, e=1,
De = i
i {T false}, e=0,
i
we associate its likelihood, given the event that the birth date is x ∈ X, with the
expression2
P(x | De) (cid:26)p (x)/p (x), e=1,
V(De |x)= i = i 0 (2-13)
i P(x) 1, e=0,
with i ∈{1,...,k} and p ≡ P . In this way, the posterior probability P(De |x)
0 0 i
(the probability of De in light of the event that the subject was born in the year
i
x ∈ X) is given by the product of V(De | x) with the prior probability P(De),
i i
accordingtoBayes’sformula.
Ifwenowconsidertwopairsof“hypotheses” Dei and Dej,whichweassume
i j
conditionallyindependent(withoutbeingnecessarilyindependent),thatis,
P(Dei,Dej |x)= P(Dei |x)P(Dej |x), e ,e ∈{0,1},
i j i j i j
thenwefind
P(x | Dei,Dej) P(Dei,Dej |x) P(Dei |x)P(Dej |x)
P(Dei,Dej |x)= i j = i j = i j
i j P(x) P(Dei,Dej) P(Dei,Dej)
i j i j
P(Dei)P(Dej)
= i j ·V(Dei |x)V(Dej |x).
P(Dei,Dej) i j
i j
More generally, given k testimonies T , to each of which there corresponds the
i
pair of events De, and given a word σ ∈ {0,1}k, if we assume the conditional
i s
independenceoftheevents(Dσs(1),...,Dσs(k)),weget
1 k
k
V(Dσs(1),...,Dσs(k)|x)=ρ (cid:89)V(Dσs(i)|x) (2-14)
1 k s i
i=1
where
(cid:81)k P(Dσs(i))
ρ = i=1 i . (2-15)
s P(Dσs(1),...,Dσs(k))
1 k
If,inaddition,thereisgroundstoassumeunconditionalindependence,i.e.,ρ =1,
s
then (2-14) simply reduces to the product rule. Under this assumption, we can
2Herethesymbol P denoteseitherthereferencemeasure P0 oranyprobabilitymeasureon X
compatiblewithit.
DATINGHYPATIA’SBIRTH:APROBABILISTICMODEL 25
evaluatetheaveragelikelihood ofthesetofinformation(Dσs(1),...,Dσs(k))with
1 k
theexpression
k
V = 1 (cid:88)V(Dσs(1),...,Dσs(k)|x)=|X|k−1(cid:88)(cid:89) pσs(i)(x). (2-16)
s |X| 1 k i
x∈X x∈Xi=1
Comparingwith(2-11),weseethat
V
w = s . (2-17)
s
N−1
(cid:80)
V
s
s=0
Inotherwords,withinthehypothesesmadesofar,theallocationofthecoefficients
(2-11)correspondstoassigningtoeachdistribution P aweightproportionaltothe
s
averagelikelihoodofthesetofinformationfromwhichitisconstructed.
3. ApplicationtoHypatia
Thismethodisnowappliedtoaparticulardatingprocess,theoneofHypatia’sbirth.
Thischoicestemsfromthedesiretostudyacasebotheasytohandleandpotentially
usefulinitsresults. TheproblemofdatingHypatia’sbirthisindeedopen,inthat
therearedifferentpossibleresolutionsoftheconstraintsimposedbytheavailable
data. According to the reconstruction given by Deakin [2007, p. 51], “Hypatia’s
birth has been placed as early as 350 and as late as 375. Most authors settle for
‘around370’”. Therearenotmanytestimonies(historicalrecords)concerningthe
birthofthe Alexandrian scientist(farmoreareabout herinfamousdeath),butthey
havethedesirablefeatureofbeingindependentofoneanother,aswillbeapparent
inthesequel,sothattheschemediscussedintheprevioussectioncanbedirectly
applied. The hope is to obtain something that is qualitatively significant when
compared to the preexisting proposals, based on a qualitative discussion of the
sources, and quantitatively unambiguous. A probability distribution for the year
of Hypatia’s birth is extracted from each testimony, the specific reasoning being
brieflydiscussedin eachcase. Eventually alldistributions arecombinedaccording
tothecriteriaoutlinedintheprevioussection.
3.1. Hypatia was at her peak between 395 and 408. Under the entry ῾Υπατία,
the Suda (a Byzantine lexicon) informs us that she flourished under the emperor
Arcadius(ἤκμασεν ἐπί τῆς βασιλείας ᾿Αρκαδίου).3
It is well established that Arcadius, the first ruler of the Byzantine Empire,
reigned from 395 to 408. Guessing an age or age interval based on the Greek
ἤκμασεν, however, is less straightforward. The word is related to ἀκμή, ‘peak’,
3ϒ166.Seehttp://www.stoa.org/sol-bin/search.pl?field=adlerhw_gr&searchstr=upsilon,166.
26 CANIOBENEDETTO,STEFANOISOLAANDLUCIORUSSO
0.2
uit
or
fls 0.15
a’
ati
p
y
H 0.1
of
y
abilit 0.05
b
o
Pr
0
35 40 45
Age
Figure 1. The probability distribution f(x) assumed associated
withone’speakyears.
0.08
h
birt
s 0.06
a’
ati
p
y
H 0.04
of
y
abilit 0.02
b
o
Pr
0
350 354 359 364 369 373
Year
Figure2. The probability distribution ϒ (ξ) for Hypatia’s birth
f
basedonherpeakyears.
and we follow the rule of thumb, going back to Antiquity, that it refers to the
periodofone’slifearound40yearsofage. Specifically,weadopttheprobability
distribution f(x) in Figure 1 to model how old Hypatia would have been at her
“peak”inArcadius’reign.
Figure2showsϒ (ξ),theprobabilitydistributionfortheyearofHypatia’sbirth
f
deduced fromthis historicaldatum; it isobtained byaveraging fourteen copiesof
thetriangular f(x),eachcenteredaroundoneoftheyearsfrom355through368—
the beginning and end points of Arcadius’s empire, shifted back by the 40 years
correspondingtothepeakof f(x).
3.2. Hypatiawasintellectuallyactivein415. ThesourcesascribeHypatia’smar-
tyrdom at the hands of a mob of Christian fanatics to the envy that many felt
on account of her extraordinary intelligence, freedom of thought, and political
influence,beingawoman. HerentryintheSuda,alreadymentioned,states:
DATINGHYPATIA’SBIRTH:APROBABILISTICMODEL 27
Τοῦτο δὲ πέπονθε διὰ φθόνον καὶ τὴν ὑπερβάλλουσαν σοφίαν, καὶ
μάλιστα εἰς τὰ περὶ ἀστρονομίαν.4
SocratesScholasticus,inhisΕκκλησιαστική Ιστορία,reports:
On accountof the self-possession andease of manner, whichshe had ac-
quiredinconsequenceofthecultivationofhermind,shenotinfrequently
appeared in public in presence of the magistrates. Neither did she feel
abashed in coming to an assembly of men. For all men on account of
herextraordinarydignityandvirtueadmiredherthemore. Yetevenshe
fellavictim tothepoliticaljealousy whichatthattime prevailed. For as
she hadfrequent interviews with Orestes,it was calumniously reported
among the Christian populace, that it was she who prevented Orestes
frombeingreconciledtothebishop.5
Becauseoftheseandsimilartestimonies,itseemsreasonabletomark415asayear
ofintellectualactivityinHypatia’slife.
To get from this information a probability distribution for the year of birth, it
isnecessarytohavetheprobabilitydistributionofbeingintellectuallyactiveata
givenage. Thiscanbecalculatedgiventheprobabilityofbeingaliveatanygiven
ageandofbeingactiveatanygivenage(ifalive),bysimplemultiplication.
To derive the first of these probability distributions we have used data from a
1974mortalitytableforItalianmales,6 clippingoff agesunder18sincethesubject
wasknown tobeintellectuallyactive. The resulting probabilitydistribution,a(x),
isshowninFigure3.
1
e
v
ali
g 0.8
n
ei
b
atia 0.6
p
y
H
of 0.4
y
bilit
a
b 0.2
o
Pr
0
18 104
Age
Figure3. Theprobabilitydistributiona(x)foranadulttoreach
agivenage. Thelifeexpectancycomesto71.8years.
4Shesufferedthis[violentdeath]becauseoftheenvyforherextraordinarywisdom,especiallyin
thefieldofastronomy.
5BookVII,Chapter15;translationfrom[SocratesScholasticus,p.160].
6Alldataaretakenfromhttp://www.mortality.org.