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From coincidences to discoveries 1
Running head: From coincidences to discoveries
From mere coincidences to meaningful discoveries
Thomas L. Griffiths
Department of Cognitive and Linguistic Sciences
Brown University
Joshua B. Tenenbaum
Department of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Address for correspondence:
Tom Griffiths
Department of Cognitive and Linguistic Sciences
Brown University, Box 1978
Providence, RI 02912
e-mail: tom griffiths@brown.edu
phone: (401) 863 9563
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From coincidences to discoveries 2
Abstract
People’s reactions to coincidences are often cited as an illustration of the irrationality of human
reasoning about chance. We argue that coincidences may be better understood in terms of rational
statistical inference, based on their functional role in processes of causal discovery and theory
revision. We present a formal definition of coincidences in the context of a Bayesian framework for
causal induction: a coincidence is an event that provides support for an alternative to a currently
favored causal theory, but not necessarily enough support to accept that alternative in light of its
low prior probability. We test the qualitative and quantitative predictions of this account through a
series of experiments that examine the transition from coincidence to evidence, the correspondence
between the strength of coincidences and the statistical support for causal structure, and the
relationship between causes and coincidences. Our results indicate that people can accurately
assess the strength of coincidences, suggesting that irrational conclusions drawn from coincidences
are the consequence of overestimation of the plausibility of novel causal forces. We discuss the
implications of our account for understanding the role of coincidences in theory change.
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From coincidences to discoveries 3
From mere coincidences to meaningful discoveries
In the last days of August in 1854, the city of London was hit by an unusually violent
outbreak of cholera. More than 500 people died over the next fortnight, most of them in a small
region in Soho. On September 3, this epidemic caught the attention of John Snow, a physician who
had recently begun to argue against the widespread notion that cholera was transmitted by bad air.
Snow immediately suspected a water pump on Broad Street as the cause, but could find little
evidence of contamination. However, on collecting information about the locations of the cholera
victims, he discovered that they were tightly clustered around the pump. This suspicious
coincidence hardened his convictions, and the pump handle was removed. The disease did not
spread any further, furthering Snow’s (1855) argument that cholera was caused by infected water.
Observing clusters of events in the streets of London does not always result in important
discoveries. Towards the end of World War II, London came under bombardment by German V-1
and V-2 flying bombs. It was widespread popular belief that these bombs were landing in clusters,
with an unusual number of bombs landing on the poorer parts of the city (Johnson, 1981). After
the war, R. D. Clarke of the Prudential Assurance Company set out to ‘apply a statistical test to
discover whether any support could be found for this allegation’ (Clarke, 1946, p. 481). Clarke
examined 144 square miles of south London, in which 537 bombs had fallen. He divided this area
into small squares and counted the number of bombs falling in each square. If the bombs fell
uniformly over this area, then these counts should conform to the Poisson distribution. Clarke
found that this was indeed the case, and concluded that his result ‘lends no support to the clustering
hypothesis’ (1946, p. 481), implying that people had been misled by their intuitions.1
Taken together, the suspicious coincidence noticed by John Snow and the mere coincidence
that fooled the citizens of London present what seems to be a paradox for theories of human
reasoning. How can coincidences simultaneously be the source of both important scientific
discoveries and widespread false beliefs? Previous research has tended to focus on only one of
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From coincidences to discoveries 4
these two faces of coincidences. Inspired by examples similar to that of Snow,2one approach has
focused on conceptual analyses or quantitative measures of coincidences that explicate their role in
rational inference (Horwich, 1982; Schlesinger, 1991), causal discovery (Owens, 1992) and
scientific argument (Hacking, 1983). An alternative approach, inspired by examples like the
bombing of London,3has analyzed the sense of coincidence as a prime example of shortcomings
in human understanding of chance and statistical inference (Diaconis & Mosteller, 1989; Fisher,
1937; Gilovich, 1993; Plous, 1993). Neither of these two traditions has attempted to explain how
the same cognitive phenomenon can simultaneously be the force driving human reasoning to both
its greatest heights, in scientific discovery, and its lowest depths, in superstition and other abiding
irrationalities.
In this paper, we develop a framework for understanding coincidences as a functional
element of causal discovery. Scientific knowledge is expanded and revised through the discovery
of causal relationships that enrich or invalidate existing theories. Intuitive knowledge can also be
described in terms of domain theories with structures that are analogous to scientific theories in
important respects (Carey, 1985; Gopnik & Meltzoff, 1997; Karmiloff-Smith, 1988; Keil, 1989;
Murphy & Medin, 1985), and these intuitive theories are grown, elaborated and revised in large
part through processes of causal discovery (Gopnik, Glymour, Sobel, Schulz, Kushnir, & Danks,
2004; Tenenbaum, Griffiths, & Niyogi, in press). We will argue that coincidences play a crucial
role in the development of both scientific and intuitive theories, as events that provide support for a
low-probability alternative to a currently favored causal theory. This definition can be made precise
using the mathematics of statistical inference. We use the formal language of causal graphical
models (Pearl, 2000; Spirtes, Glymour, & Schienes, 1993) to characterize relevant aspects of
intuitive causal theories, and the tools of Bayesian statistics to propose a measure of evidential
support for alternative causal theories that can be identified with the strength of a coincidence. This
approach allows us to clarify the relationship between coincidences and theory change, and to
make quantitative predictions about the strength of coincidences that can be compared with human
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From coincidences to discoveries 5
judgments.
The plan of the paper is as follows. Before presenting our account, we first critique the
common view of coincidences as simply unlikely events. This analysis of coincidences is simple
and widespread, but ultimately inadequate because it fails to recognize the importance of
alternative theories in determining what constitutes a coincidence. We then present a formal
analysis of the computational problem underlying causal induction, and use this analysis to show
how coincidences may be viewed as events that provide strong but not necessarily sufficient
evidence for an alternative to a current theory. After conducting an experimental test of the
qualitative predictions of this account, we use it to make quantitative predictions about the strength
of coincidences in some of the complex settings where classic examples of coincidences occur:
coincidences in space, as in the examples of John Snow and the bombing of London, and
coincidences in date, as in the famous “birthday problem”. We conclude by returning to the
paradox of coincidences identified above, considering why coincidences often lead people astray
and discussing their involvement in theory change.
Coincidences are not just unlikely events
Upon experiencing a coincidence, many people react by thinking something like ‘Wow!
What are the chances of that?’ (e.g., Falk, 1981-1982). Subjectively, coincidences are unlikely
events: we interpret our surprise at their occurrence as indicating that they have low probability. In
fact, it is often assumed that being surprising and having low probability are equivalent: the
mathematician Littlewood (1953) suggested that events having a probability of one in a million be
considered surprising, and many psychologists make this assumption at least implicitly (e.g.,
Slovic & Fischhoff, 1977). The notion that coincidences are unlikely events pervades literature
addressing the topic, irrespective of its origin. This belief is expressed in books on spirituality
(‘Regardless of the details of a particular coincidence, we sense that it is too unlikely to have been
the result of luck or mere chance,’ Redfield, 1998, p. 14), popular books on the mathematical basis
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From coincidences to discoveries 6
of everyday life (‘It is an event which seems so unlikely that it is worth telling a story about,’
Eastaway & Wyndham, 1998, p. 48), and even the statisticians Diaconis and Mosteller (1989)
considered the definition ‘a coincidence is a rare event,’ but rejected it on the grounds that ‘this
includes too much to permit careful study’ (p. 853).
The most basic version of the idea that coincidences are unlikely events refers only to the
probability of a single event. Thus, some data, d, might be considered a coincidence if the
probability of d occurring by chance is small.4On September 11, 2002, exactly one year after
terrorists destroyed the World Trade Center in Manhattan, the New York State Lottery “Pick 3”
competition, in which three numbers from 0-9 are chosen at random, produced the results 9-1-1
(Associated Press, September 12, 2002). This seems like a coincidence,5and has reasonably low
probability: the three digits were uniformly distributed between 0 and 9, so the probability of such
a combination is (1
10)3or 1 in 1000. If d is a sequence of ten coinflips that are all heads, which we
will denote HHHHHHHHHH, then its probability under a fair coin is (1
2)10or 1 in 1024. If d is an
event in which one goes to a party and meets four people, all of whom are born on August 3, and
we assume birthdays are uniformly distributed, then the probability of this event is (1
365)4, or 1 in
17,748,900,625. Consistent with the idea that coincidences are unlikely events, these values are all
quite small.
The fundamental problem with this account is that while coincidences must in general be
unlikely events, there are many unlikely events that are not coincidences. It is easy to find events
that have the same probability, yet differ in whether we consider them a coincidence. In particular,
all of the examples cited above were analyzed as outcomes of uniform generating processes, and so
their low probability would be matched by any outcomes of the same processes with the same
number of observations. For instance, a fair coin is no more or less likely to produce the outcome
HHTHTTHTHT as the outcome HHHHHHHHHH. Likewise, observing the lottery numbers 7-2-3
on September 11 would be no more likely than observing 9-1-1, and meeting people with birthdays
on May 14, July 8, August 21, and October 4, would be just as unlikely as any other sequence,
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From coincidences to discoveries 7
including August 3, August 3, August 3, and August 3. Using several other examples of this kind,
Teigen and Keren (2003) provided empirical evidence in behavioral judgments for the weak
relationship between the surprisingness of events and their probability. For our purposes, these
examples are sufficient to establish that our sense of coincidence is not merely a result of low
probability.
We will argue that coincidences are not just unlikely events, but rather events that are less
likely under our currently favored theory of how the world works than under an alternative theory.
The September 11 lottery results, meeting four people with the same birthday, and flipping ten
heads in a row all grab our attention because they suggest the existence of hidden causal structure
in contexts where our current understanding would suggest no such structure should exist. Before
we explore this hypothesis in detail, we should rule out a more sophisticated version of the idea
that coincidences are unlikely events. The key innovation behind this definition is to move from
evaluating the probability of a single event to the probability of an event of a certain “kind”, with
coincidences being events of unlikely kinds. Hints of this view appear in experiments on
coincidences conducted by Falk (1989), who suggested that people are ‘sensitive to the extension
of the judged event’ (p. 489) when evaluating coincidences. Falk (1981-1982) also suggested that
when one hears a story about a coincidence, ‘One is probably not encoding the story with all its
specific details as told, but rather as a more general event “of that kind” ’ (p. 23). Similar ideas
have been proposed by psychologists studying figural goodness and subjective randomness (e.g.,
Garner, 1970; Kubovy & Gilden, 1991), and such an account was worked out in detail by
Schlesinger (1991), who explicitly considered coincidences in birthdays. Under this view, meeting
four people all born on August 3 is a bigger coincidence than meeting those born on May 14, July
8, August 21, and October 4 because the former is of the kind all on the same day while the latter is
of the kind all on different days. Similarly, the sequence of coinflips HHHHHHHHHH is more of a
coincidence than the sequence HHTHTTHTHT because the former is of the kind all outcomes the
same while the latter is of the kind equal number of heads and tails; out of all 1024 sequences of
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From coincidences to discoveries 8
length 10, only two are of the former kind, while there are 252 of the latter kind.
The “unlikely kinds” definition runs into several difficulties. First there are the problems of
specifying what might count as a kind of event, and which kind should be used when more than
one is applicable. Like the coinflip sequence HHTHTTHTHT, the alternating sequence
HTHTHTHTHT falls under the kind equal number of heads and tails, but it appears to present
something of a coincidence while the former sequence does not. The “unlikely kinds” theory
might explain this by saying that HTHTHTHTHT is also a member of a different kind, alternating
heads and tails, containing only two sequences out of the possible 1024. But why should this
second kind dominate? Intuitively, the fact that it is more specific seems important, but why? And
why isn’t alternation as much of a coincidence as repetition, even though the kinds all outcomes
the same and alternating heads and tails are equally specific? How would we assess the degree of
coincidence for the sequence HHHHHHHTTT? It appears more coincidental than a merely
“random” sequence like HHTHTTHTHT, but what “kind of event” is relevant? Finally, why do we
not consider a kind like all outcomes that begin HHTHTTHTHT..., which would predict that the
sequence HHTHTTHTHT is in fact the most coincidental of all? The situation becomes even more
complex when we go beyond discrete events. For example, the bombing of London suggested a
coincidence based upon bomb locations, which are not easily classified into kinds.
For the “unlikely kinds” definition to work, we need to be able to identify the kinds relevant
to any contexts, including those involving continuous stimuli. The difficulty of doing this is a
consequence of not recognizing the role of alternative theories in determining what constitutes a
coincidence. The fact that certain kinds of events seem natural is a consequence of the
theory-ladenness of the observer: there is no a priori reason why any set of kinds should be favored
over any other. In the cases where definitions in terms of unlikely kinds do seem to work, it is
because the kinds being used implicitly correspond to the predictions of a reasonable set of
alternative theories. To return to the coinflipping example, kinds defined in terms of the number of
heads in a sequence implicitly correspond to considering a set of alternative theories that differ in
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From coincidences to discoveries, Figure 5
h0: uniform
h1: uniform+regularity
RR
?C
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From coincidences to discoveries, Figure 6
A
Change in...
Number
BC
4
6
8
Human dataBayesian model
D
Ratio
BE
4
6
8
G
Location
FB
4
6
8
H
Spread
BI
4
6
8
J
(Uniform)
KL
4
6
8
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From coincidences to discoveries, Figure 7
4
BB
PPPPPP654321
12
PPPPPP
321465
123
56
BBBB
3456
BBBBBB
P(Graph 0|h0) = 1
P(Graph 0|h1) = (1 − p)6
P(Graph 60|h0) = 0
P(Graph 60|h1) = p4(1 − p)2
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From coincidences to discoveries, Figure 8
h1: uniform+regularity
P(bi)
1365
B = August
h0: uniform
P(bi)
1365
DayDay
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From coincidences to discoveries, Figure 9
05 10
Feb 25, Aug 10 (N)
Feb 11, Apr 6, Jun 24, Sep 17 (M)
Jan 23, Feb 2, Apr 9, Jul 12, Oct 17, Dec 5 (L)
Feb 22, Mar 6, May 2, Jun 13, Jul 27, Sep 21, Oct 18, Dec 11 (K)
May 18, May 18 (J)
Sep 30, Oct 1 (I)
Aug 3, Aug 3, Aug 3, Aug 3 (H)
Jun 27, Jun 29, Jul 1, Jul 2 (G)
Jan 2, Jan 13, Jan 21, Jan 30 (F)
Jan 17, Apr 17, Jun 17, Nov 17 (E)
Jan 29, Apr 26, May 5, May 5, May 5, May 5, Sep 14, Nov 1 (D)
Jan 12, Mar 22, Mar 22, Jul 19, Oct 1, Dec 8 (C)
Feb 12, Apr 6, May 6, Jun 27, Aug 6, Oct 6, Nov 15, Dec 22 (B)
Mar 12, Apr 28, Apr 30, May 2, May 4, Aug 18 (A)
How big a coincidence?
Human data
0510
Statistical evidence
Bayesian model
0510
Statistical evidence
Without sizes
0510
Statistical evidence
Uniform P(B)
0510
Statistical evidence
Unit weights
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From coincidences to discoveries, Figure 10
123456789
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
B
A
C
D
E
FG
I
H
J
KL
Bombing (coincidence)
Lemurs (cause)
r = 0.995
ρ = 0.993
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From coincidences to discoveries, Figure 11
2345
Packages (cause)
6789 10
1
2
3
4
5
6
7
8
9
10
N
M
L
K
H
J
D
C
I
G
F
A
E
B
Birthdays (coincidence)
r = 0.927
ρ = 0.903