Bayesian Thought in Early Modern Detective Stories: Monsieur Lecoq, C. Auguste Dupin and Sherlock Holmes

7 Pages·0.11 MB·English

by  Joseph B. Kadane

StatisticalScience
2009,Vol.24,No.2,238–243
DOI:10.1214/09-STS298
(cid:13)c InstituteofMathematicalStatistics,2009
Bayesian Thought in Early Modern
Detective Stories: Monsieur Lecoq,
C. Auguste Dupin and Sherlock Holmes
Joseph B. Kadane
0
1
0
2
Abstract. This paper reviews the maxims used by three early mod-

n ern fictional detectives: Monsieur Lecoq, C. Auguste Dupin and Sher-
a
J lock Holmes. It find similarities between these maxims and Bayesian
9 thought. Poe’s Dupin uses ideas very similar to Bayesian game theory.
1 Sherlock Holmes’ statements also show thought patterns justifiable in

Bayesian terms.
]
E
Key words and phrases: ArthurConanDoyle,EdgarAllanPoe,Emile
M
Gaboriau, odd and even, Bayesian Game Theory.
.
t
a
t
s 1. INTRODUCTION think.Althoughindirectmethodssuchas functional
[
magnetic resonance imaging (fMRI) cannot yet be
The three writers considered here, Emile Gabo-
1 used to tell what a person is thinking, it is not im-
riau (1832–1873), Edgar Allan Poe (1809–1849) and
v possible that in the future this may be possible. We
3 Sir Arthur Conan Doyle (1859–1930) are considered
have so-called lie-detector machines, although an-
5 to be the founders of the modern interest in detec-
2 other National Research Council Report (2003) has
tivestories,writingevenbeforetheterm“detective”
3 seriously challenged their accuracy.
. was used (Bleiler, 1975). In addition to the many
1 On the other hand, there is a sense in which un-
novels and short stories that have ensued, there are
0 derstandingfictionalcharactersiseasierthanunder-
0 also popular television crime, mystery and police
standing real ones. There is a fixed body of written
1 shows that can be considered progeny. This pop-
: work, and this is all the evidence there will ever be.
v ularity continues despite the weaknesses in the un-
Those words tell what characteristics of the detec-
i derlying forensic science, as emphasized by a recent
X tives the author considers most important. When
report of the National Research Council (2009).
r the author writes about the way such characters
a Thisessayaimstoexaminethepatternofthought
go about their work, this can be taken to be au-
usedby their respective detectives: MonsieurLecoq,
thoritative. Thus there is no issue, for example, of
C. Auguste Dupin and Sherlock Holmes. What does
changingormoldingtheresponsesthatarealperson
it mean to understand the thoughts of a fictional
mightgiveinresponsetoaquestionframedinaway
character? With a real person, one can ask ques-
the respondent hadn’t considered previously. Thus
tions and run experiments to ascertain how they
we may take the statements written by the authors
as summaries of what they intend their characters
Joseph B. Kadane is Leonard J. Savage University
(here, detectives) to teach the readers.
Professor of Statistics and Social Sciences, Emeritus,
Department of Statistics, Carnegie Mellon University,
2. EMILE GABORIAU’S MONSIEUR LECOQ
Pittsburgh, Pennsylvania 15223, USA
e-mail: [email protected]. The plot of the novel Monsieur Lecoq (1869) and
itssequelInHonor of the Name Gaboriau(1975)re-
This is an electronic reprint of the original article
volvesaroundtheeffortsofnovicedetectiveLecoqto
published by the Institute of Mathematical Statistics in
Statistical Science, 2009, Vol. 24, No. 2, 238–243. This establish the identity of a prisoner who killed three
reprint differs from the original in pagination and people in a brawl in a bar on the outskirts of Paris.
typographic detail. The intricacies are enormous, and in the end Lecoq
1
2
J. B. KADANE
goes to a wise-man consulting detective who essen- hiding places in Minister D.’s rooms, behind the
tially solves the case for him. wallpaper, hidden in a hollow leg of furniture, etc.,
Little is written about Lecoq’s methods, except without result. They have also twice found ways to
foronerefrain thatoccurs threetimes:“Always sus- search Minister D.’s body,again withoutfindingthe
pect that which seems probable; and begin by be- letter.
lieving that which appears incredible” (page 79), What is important to us in these three stories
“Distrust all circumstances that seem to favor your is the theory Poe promulgates as to how Dupin is
secretwishes”(page 87), and“Always distrustwhat thinking about the puzzles he sets himself to solve.
seems probable!” (page 248). Taken together they In a preamble to The Murders in the Rue Morgue,
suggest a tinge of paranoia, perhaps (but what is a Poe writes of his views on skill in games. The first
poor detective to do in the hands of a malign au- of these is chess, which he regards as principally a
thor intent on surprising the reader and, one sup- matter ofattention (theloser,byinattention, makes
poses, his own detective?). It also suggests a touch a blunder). (I don’t think Poe is correct about chess
of Lindley’s Cromwell’s Rule,notto putzeroproba-
among decent players, as winners are often those
bility on any conceivable possibility (Lindley, 1985,
who employ sound openings, develop their pieces,
page 104).
pay attention to their pawn structure, protect their
We can’t go further with Lecoq’s maxims or the-
king, fight for control of important center squares,
ories, because Gaboriau doesn’t give us any.
and gradually accumulate little advantages all the
while thwarting his opponent’s attempts to do the
3. EDGAR ALLAN POE’S C. AUGUSTE DUPIN
same to him.) Draughts (now called checkers) inter-
In The Murders in the Rue Morgue Poe (1944), ests Poe more. For example, if the game is reduced
Dupin and a friend read newspaper accounts of two to four kings, he writes “Deprived of ordinary re-
murders in a fourth story room locked from the in- sources, the analyst throws himself into the spirit of
side. The mother’s throat was slashed many times; his opponent, identifies himself therewith, and not
the daughter was suffocated and her body stuffed unfrequently sees thus, at a glance, the sole meth-
up a chimney. Dupin offers to help the police, and ods (sometimes absurdly simple ones) by which he
is given access to the crime scene. He findshair that may seduce into error or hurry into miscalculation”
he is sure is not human. (I won’t leave you hang- (page 47).
ing too long, but first want to introduce you to the The third game to interest Poe is whist, which is
problems in the other two Dupin stories.) roughly like bridge without bidding.
In The Mystery of Marie Rogˆet, a young woman’s
...proficiencyinwhistimpliescapacityfor
body is found floating in the Seine River. Using
successinallthesemoreimportantunder-
newspaper accounts, Dupin challenges much of the
takings where mind struggles with mind.
rationale in those stories. The case and the news-
When I say proficiency, I mean that per-
paper accounts all did occur in New York City, to
fection in the game which includes a com-
a young woman named Mary Cecilia Rogers. Her
prehension of all the sources whence le-
case is still regarded as unsolved, although there are
gitimateadvantagemaybederived.These
hints that her death may have been the result of an
unsuccessful abortion. arenotonly manifold,butmultiform,and
Finally, The Purloined Letter is the most famous lie frequently among recesses of thought
of the three Dupin stories. A police prefect asks altogether inaccessibletotheordinaryun-
Dupin’s help in finding and returning a letter con- derstanding.
cerning a high-placed lady who saw the letter being ...it is in matters beyond the limits of
takenbyMinisterD.,butwashelplesstopreventthe mere rule that the skill of the analyst is
theft. Minister D. has since been using the letter for evinced. He makes, in silence, a host of
blackmail. The prefect states that the letter has not observations and inferences. So, perhaps,
been revealed, because the consequences that would do his companions; and the difference in
have ensued from its release have not occurred. Sec- the extent of the information obtained,
ond, Minister D. must have the letter close at hand lies not so much in the validity of the in-
for it to beusefulto him.Finally, usingvarious sub- ference as in the quality of the observa-
terfuges, the police have carefully searched all the tion. The necessary knowledge is that of
3
BAYESIANDETECTIVES
what to observe. Our player confines him- could fasten itself. He thus establishes how the mur-
self not at all; nor, because the game is derers escaped.
the object, does hereject deductions from Putting together the wild brutality of the mur-
things external to the game. He exam- ders, the enormous strength it must have taken to
ines the countenance of his partner, com- put the dead daughter’s body up the chimney, and
paring it carefully with that of each of the nonhuman hair he found on the scene, Dupin
his opponents. He considers the mode of concludesthatsomeape-likeanimalmusthavedone
assorting the cards in each hand; often thesemurders.Heplacesanadaskingifsomeonehas
counting trump by trump, and honor by lost an “Ourang-Outang.” When a sailor shows up,
honor, through the glances bestowed by Dupin learns that he had an orangutan which es-
their holds on each. He notes every varia- caped with the sailor’s shaving razor, murdered the
tion of face as the lay progresses, gather- two women, and escaped.
ing a fund of thought from the differences What can we say of how well Dupin’s deductions
in the expression of certainty, of surprise,
used Poe’s theories? Certainly his deduction about
of triumph, of chagrin. From the manner
egress is using a combination of Bayes’ rule and
of gathering up a trick he judges whether
Lindley’s “Cromwell’s Rule.” However, perhaps we
the person taking it, can make another in
canexcusethisomissionoftheuseofhisideasabout
the suit.
game theory in the light of the bouquet Poe throws
The first two or three rounds having been
to all the readers of this journal:
played, he is in full possession of the con-
tentsofeachhand,andthenceforwardputs Coincidences, in general, are great stum-
down his cards with as absolute a preci- bling-blocks in the way of that class of
sion of purpose as if the rest of the party thinkers who have been educated to know
hadturnedoutwardthefacesoftheirown. nothing of the theory of probabilities—
The analytical power would not be con- that theory to which the most glorious of
founded with simple ingenuity; for while human research are indebted for the most
the analyst is necessarily ingenious, the glorious of illustration (page 77).
ingenious man is often remarkably inca-
Asalreadymentioned,TheMysteryofMarieRogˆet,
pable of analysis.
remains a mystery. Solving real cases is no doubt
Between ingenuity and the analytic abil-
moredemandingthansolvingfictionalones.Nonethe-
ity there exists a difference far greater,
less, there is one passage that demands our atten-
indeed, than that between the fancy and
tion. Poe writes,
the imagination, but of a character very
strictlyanalogous.Itwillbefound,infact, [The journal L’Etoile writes] ‘All experi-
thattheingeniousarealwaysfanciful,and
ence has shown that drowned bodies, or
thetruly imaginativeneverotherwisethan
bodiesthrownintothewater immediately
analytic (pages 48, 49).
after death by violence, require from six
Itisthistalent thatDupiniscalled upontoexem- to ten days for sufficientdecomposition to
plify.For example, howdidthemurdererormurder- take place to bring them to the top of the
ers escape from the murder room in The Murders in water.’
the Rue Morgue? “I knew that all apparent impos- Theseassertionshavebeentacitlyreceived
sibilities must be proved not to be such in reality. I by every paper in Paris, with the excep-
1
proceededto think,thus—aposteriori. Themurder- tion of Le Moniteur. Thislatter printen-
ersdid escapefromoneofthesewindows.Thisbeing deavorstocombatthatportionofthepara-
so, they could not have re-fastened the sashes from graph which has reference to ‘drowned
the inside, as they were found to be fastened...Yet bodies’ only, by citing some five or six in-
the sashes were fastened. They must, then, have the stances in which the bodies of individuals
power of fastening themselves” (pages 72, 73, em-
phasis in original). Dupin then goes on to show that 1The New York Commercial Advertiser, edited by Col.
while the two windows look the same, one of them Stone.
4
J. B. KADANE
known to be drowned were found float- asks, ‘Are they even or odd?’ Our school-
ing after the lapse of less time than is in- boyreplies,‘Odd,’andloses;butuponthe
sisteduponbyL’Etoile.Butthereissome- second trial he wins, for he then says to
thing excessively unphilosophical in the himself: “The simpleton had them even
attempt,onthepartofLeMoniteur,tore- uponthefirsttrial,andhisamountofcun-
but the general assertion of L’Etoile, by a ning is just sufficient to make him have
citation of particular instances militating themodduponthesecond;Iwilltherefore
against that assertion. Had it been possi- guess odd’;—he guesses odd, and wins.
bletoadducefiftyinsteadoffiveexamples Now, with a simpleton a degree above the
of bodies found floating at the end of two first, he would have reasoned thus: ‘This
or three days, these fifty examples could fellowfindsthatinthefirstinstanceIguessed
still have been properly regarded only as odd,and,inthesecond,hewillproposeto
exceptions to L’Etoile’s rule, until such himself, upon the first impulse, a simple
time as the rule itself should be confuted. variationfromeventoodd,asdidthefirst
Admitting the rule (and this Le Moniteur simpleton; but then a second thought will
does not deny, insisting merely upon its suggest that this is too simpleavariation,
exceptions), the argument of L’Etoile is and finally he will decide upon putting
sufferedto remainin fullforce;for this ar- it even as before. I will therefore guess
gument does not pretend to involve more even’;—he guesses even, and wins. Now
than a question of the probability of the this mode of reasoning in the school-boy,
body having risen to the surface in less whom his fellows termed ‘lucky,’—what,
than three days; and this probability will in its last analysis, is it?
be in favor of L’Etoile’s position until the ‘It is merely,’ I said, ‘an identification of
instances so childishly adduced shall be thereasoner’sintellectwiththatofhisop-
sufficient in number to establish an an- ponent’ (pages 165, 166).
tagonistical rule (pages 112, 113).
If we identify utilities with marbles, this is a zero-
sum two person game. The minimax strategy, inde-
This is an important and subtle point, one that
pendently one-half probability on odds and half on
it took the medical profession another century to
evens, seems a very poor recommendation to this
incorporate, via the use of controlled clinical trials.
young genius.
Finally, we come to Poe’s masterpiece, The Pur-
Dupin reasons as follows: Minister D. knows the
loined Letter. Dupin further expands on the theory
methods of the police. They are extremely good at
he is using in solving the case by discussing yet an-
ferreting out and exploring all of the hidden places
other game, as follows:
theletter might be.None of thosesearchers has suc-
I knew one [school-boy] about eight years ceeded. Thus Minister D., knowing what he does
of age, whose success at guessing in the about the police, does not put it in any of these
game of ‘even and odd’ attracted univer- places. Where then? It must be in plain sight!
sal admiration. This game is simple, and To test hisidea, Dupin visits Minister D., wearing
is played with marbles. One player holds eyeglasses that obscure where his eyes are focusing.
in his hand a number of these toys and He sees a scruffy document hanging, observes that
demands of another whether that number it is folded inside out, and deduces that this must
is even or odd. If the guess is right, the be the letter. Leaving a gold snuff box so he has
guesser wins one; if wrong, he loses one. an excuse to return the next day, Dupin has a doc-
The boy to whom I allude won all the ument prepared that matches the new exterior of
marbles of the school. Of course he had the letter. Returning the next day, purportedly to
some principle of guessing; and this lay pick up his snuff box, a cannon goes off in the street
in mere observation and admeasurement below (which Dupin had arranged). Minister D. is
of the astuteness of his opponents. For distracted, Dupin switches his fake for the real let-
example, an arrant simpleton is his op- ter, and leaves with both the real letter and his gold
ponent, and, holding up his closed hand, snuff box.
5
BAYESIANDETECTIVES
Inthisstory,Poehasfinallydeliveredonhispromise.analytic reasoning: “Most people, if you describe a
Dupin has used his understanding of Minister D.’s train of events to them, will tell you what the result
thought process to identify where the letter is. The would be. They can put those events together in
Purloined Letter is a wonderful story. It enchanted their minds, and argue from them that something
me as a child, and still does. will come to pass. There are few people, however,
who, if you told them a result, would be able to
4. SIR ARTHUR CONAN DOYLE’S evolve from their own inner consciousness what the
SHERLOCK HOLMES stepswerethatledtothatresult.Thispoweriswhat
ImeanwhenItalkofreasoningbackward,oranalyt-
Wedonotneedtospeculateabouttheantecedents
ically” (pages 83, 84). So Doyle (Holmes) is saying
Doyle had in mind. In his autobiography, Memories
that predicting subsequent from preceding events is
and Adventures Doyle (1924) he writes:
relatively straightforward, but the reverse is hard.
Gaboriau had rather attracted me by the And this is exactly what Bayes’ Theorem does.
neat dovetailing of his plots, and Poe’s However, that theorem is even more evident in
masterful detective, M. Dupin, had from what we must take as Holmes’ slogan, as it is re-
boyhoodbeenoneofmyheroes.Butcould peated four times in the work (pages 111, 315, 926,
I bring an addition of my own? I thought 1011). “When you have eliminated the impossible,
of my old teacher Joe Bell, ...of his eerie whatever remains, however unlikely, must be the
truth.”
trick of spotting details. If he were a de-
This is formulated sufficiently crisply that it can
tective, he would surely reduce this fasci-
nating but unorganized business to some- actually be proved as follows: Let H1,...,Hk be k
theories of the case, mutually exclusive (not more
thing nearer an exact science (page 69).
thanonecanbetrue),exhaustive(oneofthemmust
In contrast to Gaboriau’s single (or perhaps dou- betrue), and supposeeach has positive prior proba-
ble)bookandPoe’sthreeshortstories,Doyle(1981) bility. In fact, we’ll think of H1 as the theory (how-
gives us four Sherlock Holmes novels and 56 short ever unlikely) that is not eliminated by the data.
stories. So we have in one sense a great deal of in- Suppose we have data X that has eliminated theo-
formation. However, Doyle seems less anxious than ries H2,...,Hk, that is,
Poe to show us how Holmes is thinking about his
tasks. When he does so, on occasion those thoughts (1) P{X|Hi}=0, i=2,...,k,
are often reminiscent of ideas already in Poe’s sto- but P{X|H1)6=0.
ries. For example, in The Adventure of the Second Then
Stain from The Return of Sherlock Holmes, a letter
from a foreign power has been stolen. If its content P{H1|X}
were known, it could cause various foreign upsets. P{X|H1}P{H1}
= (this is Bayes’ Theorem)
“Only one importantthing has happenedin thelast k
Pi=1P{X|Hi}P{Hi}
three days, and that is that nothing has happened”
(page 659). It seems to me that this is much like the P{X|H1}P{H1}
= substituting (1)
evidenceinThe Purloined Letter thattheletter had P{X|H1}P{H1}
not been used. =1.
Similarly, in The Hound of the Baskervilles,
Holmessays“If...wearedealingwithforcesoutside Thus no matter how small P{H1} may have been,
the posterior probability of H1 under these circum-
the ordinary laws of Nature, there is an end to our
stances is one.
investigation. Butweare boundtoexhaustall other
ThusSherlockHolmesisusing,andinsistingupon,
hypotheses before falling back upon this one” (page
Bayesian results to explain his actions.
684).IfindthisreasoningsimilartothatDupinused
in ascertaining how themurderer(s)left thescene of
5. CONCLUSION
the murders in Murders in the Rue Morgue.
Thereis onepassage, intheconclusion to A Study To make this conclusion maximally embarrassing
in Scarlet, where I think Doyle is aiming to clar- (andhencepossiblyentertainingtothereader),Ipresent
ify what Poe is trying to get at on the subject of it as if I were being cross-examined by C:
6
J. B. KADANE
C: Dr. Kadane, is it correct that you are familiar C: On the other hand, is there anything in your
with Bayes’ Theorem? paper that would have helped Poe had it been
K: Yes. available more than a century before it was?
C: How would you describeSherlock Holmes’ useof K: Nothing I can think of.
Bayesian ideas? C: Now, is it also the case that you have written on
K: Holmes certainly seems to understandthe ideas, the subject of skill in games, is that correct?
and how to use them. K: Yes.Fourofus,theotherswerePatLarkeyagain,
C: Is it known whether Doyle had an acquaintance RobertAustinandShmuelZamir,wroteapaper
with mathematics sufficient that he might be fa- by that title, published in Operations Research
miliar with a mathematical version of the theo- (Larkey et al., 1997).
rem? C: Again, briefly, what is this paper about?
K: It is known that Doyle was trained and qualified K: We create asimplified version of poker,and sim-
ulate contests among various strategies for play-
asaphysician.Idonotknowtheextenttowhich
ing the game. One interesting finding was non-
some math may have been part of that training.
transitivity: under certain circumstances, there
Thereforemyanswertoyourquestionis“Idon’t
could be strategies A, B and C, where A is ef-
know.”
fective against B, B against C and C against A.
C: Very well,whetherornotDoyle hadthatmathe-
So there isn’t among these, a “best” strategy at
matical training, does Holmes, in Doyle’s hands,
all.
correctly use Bayes’ Theorem?
C: Is there anything in this paper that would have
K: Yes, I think he does.
deepened Poe’s understanding of skill in games?
C: Does he make any errors he might have avoided
K: I don’t think so. I think Poe understood skill
had Doyle had a more mathematical grasp of
in games very well, both in how Dupin outwits
Bayes’ Theorem?
Minister D., and in his general introduction. As
K: None that I have found.
I explained earlier, I disagree with him about
C: Verywell,nowlet’sturntoPoe’swork.Dr.Kadane,
chess, but as a general matter, his view of skill
do I understand correctly that you have writ-
in games is very similar to theonein ourpapers.
tenapaperabouttheconnectionbetweengames
C: So then is it your thought that you have very
and Bayesian theory?
little news for either Doyle or Poe?
K: Yes, Pat Larkey and I wrote a paper entitled
K: Yes, I think that is fair.
“SubjectiveProbabilityandtheTheoryofGames”
C: Thenwhathas beengoing onin this fieldfor the
(Kadane and Larkey, 1982).
last 100 or 150 years? Have we gotten nowhere?
C: Would you tell us briefly the main argument of
K: Idon’tthinkthatisafaircharacterization.What
that paper?
isnewisthatthroughtheworkofRamsey(1926),
K: Surely. The idea is that if I am playing a game
de Finetti (1970, 1975), Savage (1954), DeGroot
against you, my main source of uncertainty is
(1970) and Lindley (1985), we now have a gen-
what you willdo. As a Bayesian I have probabil-
eral theory of what it means to make good de-
ities on what you will do, and can use them to
cisions in the face of uncertainty. That theory
calculate my maximum expected utility choice,
rests on a few simple principles:
which is what I should choose. – all sources of uncertainty are modeled proba-
C: Is this consistent with what Poe writes about bilistically,
games? – asdatabecameavailable,theprobabilitymod-
K: Verymuchso.ThemarblekingofPoe’sacquain- els are updated by conditioning on the ob-
tance is very good at guessing his opponent’s served data,
strategy, which is how he winds up with all the – whenitisrequiredthatdecisionsbemade,the
marbles in his school. Dupin is successful at un- optimal decision maximizes expected utility,
derstanding Minister D.’s strategy, and hence in wheretheexpectation is taken withrespectto
finding and retrieving the letter. the current (updated) opinion of the decision
C: Is there anything that Poe writes about games maker.
that is inconsistent with your theory? Thus we now understand both Bayes’ Theorem
K: No. and the Bayesian approach to games as special
7
BAYESIANDETECTIVES
cases of this very general theory. That’s what’s Gaboriau, E. (1975). Monsieur Lecoq. Dover Publications,
new. New York.
Kadane, J. and Larkey, P. (1982). Subjective probability
C: Thank you, Dr. Kadane.
and the theory of games. Management Sci. 28 113–120.
Both detective stories and Bayesian analysis have MR0652016
Larkey,P., Kadane,J.,Austin,R.andZamir,S.(1997).
flourished in the intervening century. They share
Skill in games. Management Sci. 43 596–609.
some common roots.
Lindley, D. (1985). Making Decisions. Wiley, Chichester.
MR0892099
National Research Council (2003). The polygraph and
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