Source: https://www.bayesianspectacles.org/dennis-lindleys-second-paradox/
Timestamp: 2019-04-25 20:06:22+00:00

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Specifically, in 1997 Lindley argued that Bayes factor proponents could find themselves simultaneously saying that (a) it is more likely to be a tall man than a tall woman, (b) it is more likely to be a short man than a short women, and (c) it is more likely to be a woman than a man. With his characteristic wit, he concludes that “one hardly advances the respect with which statisticians are held in society by making such declarations.” He further points out that this scenario is conceptually related to Simpson’s paradox.
In a short commentary, we illustrate Lindley’s critique for a simple example. We conclude that Lindley’s phrasing is imprecise, and that the paradoxical property he points out is actually intuitive. All told, it seems the Bayes factor is quite alright.
Dablander, F., van den Bergh, D., & Wagenmakers, E. J. (2018). Another Paradox? A Comment on Lindley (1997). PsyArXiv. November, 8.
Degroot, M. H. (1982). Comment. Journal of the American Statistical Association, 77(378), 336-339.
Lindley, D. V. (1957). A statistical paradox. Biometrika, 44(1/2), 187-192.
Lindley, D. V. (1975). The future of statistics: a Bayesian 21st century. Advances in Applied Probability, 7, 106-115.
Lindley, D. V. (1986). Comment. The American Statistician, 40(1), 6-7.
Lindley, D. V. (1997). Some comments on Bayes factors. Journal of Statistical Planning and Inference, 61(1), 181–189.
Lindley, D. V. (1991). Making Decisions. John Wiley & Sons.
Lindley, D. V. (2006). Understanding uncertainty. John Wiley & Sons.
1 As per Sigler’s law of eponymy, the “paradox” was discovered earlier by Harold Jeffreys. Bartlett pointed out an (inconsequential) error in Lindley’s exposition, and it is thus sometimes called the “Jeffreys-Lindley-Bartlett” paradox.
2 An interview with Dennis Lindley is available from here. He wrote two accessible books on probability and decision making, see here and here.
Fabian Dablander is a PhD candidate at the Psychological Methods Group of the University of Amsterdam. You can find him on Twitter @fdabl.
Don van den Bergh is a PhD candidate at the Psychological Methods Group of the University of Amsterdam.

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