Abstract
Suppose that several observations “coincide,” meaning that they are similar in some interesting respect. Is this coinciding a mere coincidence, or does it derive from a common cause? Those who reason about this kind of question—whether they embrace the first answer or the second—often deploy a mode of inference that I call probabilistic modus tollens. In this chapter I criticize probabilistic modus tollens and consider likelihood and Bayesian frameworks for reasoning about coincidences. I also consider the perspective offered by model selection theory (including the Akaike information criterion), and argue that model selection often provides important insights.
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Notes
- 1.
Go to http://www.angio.net/pi/piquery to see if your birthday appears in the first 100 million digits.
- 2.
The naïve and the sophisticated are characters in my story; I do not mean to suggest that all sophisticated thinkers in the real world reason exactly in the way I’ll describe the sophisticated as reasoning.
- 3.
Darwin (1859, ch. 13) argued that adaptive similarities between species provide poor evidence for common ancestry and that it is useless and deleterious similarities that provide more powerful evidence; see Sober (2008, 2011) for discussion. Darwin (1871, ch. 6) noticed the parallel epistemological problems that connect historical linguistics and phylogenetic inference.
- 4.
Notice also that the argument that appeals to (3) to show that the number of heads is a sufficient statistic depends on using the likelihood ratio as the relevant method for comparing the two estimates. If the likelihood difference were used instead, the corresponding equality would not be true. How one measures weight of evidence matters; see Fitelson (1999) for further discussion.
- 5.
I do not use the term “metaphysical” here in the pejorative sense sometimes used by logical positivists. Rather, my use of the term contrasts with “epistemological.” The former has to do with the way the world is, the latter with the beliefs we should form about the world.
- 6.
One reason that Reichenbach’s principle should not be formulated metaphysically is the fact that it is at least a defensible position to maintain that quantum mechanics describes event types that are lawfully correlated but not causally connected. Arthur Fine has pointed out to me that these correlations also show that my categories of Mere Coincidence and Causal Connection are not exhaustive.
- 7.
- 8.
See Griffiths and Tenenbaum (2007) for an interesting psychological study of how people actually think about coincidences that uses a Bayesian framework.
- 9.
L(FAIR) can’t have a higher likelihood than L(PL*) or L(PLT*), either.
- 10.
Cross validation makes no explicit mention of simplicity, but shares with AIC the goal of finding models that will be predictively accurate. It is interesting that there is a form of cross-validation (“take-one-out” cross validation) that is asymptotically equivalent with AIC (Stone 1977).
- 11.
It might be suggested that the hypothesis that the two lotteries were fixed to ensure that Adams would win is a hypothesis that would occur to you only after you observe Adams’ double win, and that it is a rule of scientific inference that hypotheses must be formulated before the data are gathered to test them. This temporal requirement is often invoked in frequentist statistics. For discussion, see Hitchcock and Sober (2004). It is a point in favor of the model selection approach that one does not have to invoke this temporal requirement to explain what is wrong with the PL and the PLT models.
- 12.
See Wardrop (1999) for a skeptical assessment of Gilovich et al.’s analysis. Wardrop argues that Gilovich et al. tested hypotheses about correlation (whether a player’s probability of scoring on a given shot if he scored on earlier shots is greater than his probability of scoring if he missed previously), but did not assess the issue of stationarity (maybe a player’s probability of scoring suddenly shifts from one value to another). Wardrop suggests that the latter may be the relevant consideration.
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Acknowledgements
I thank Matthew Kopec, Ellery Eells, Arthur Fine, Malcolm Forster, George Gale, Michael Goldsby, Daniel Hausman, Stephen Leeds, Wouter Meijs, David Myers, Joshua Tenenbaum, and Naftali Weinberger for helpful discussion, and Nick Harding for permitting me to reprint his cartoon.
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Sober, E. (2012). Coincidences and How to Reason About Them. In: de Regt, H., Hartmann, S., Okasha, S. (eds) EPSA Philosophy of Science: Amsterdam 2009. The European Philosophy of Science Association Proceedings, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2404-4_30
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