What is commonly referred to as “Lindley’s paradox” exposed a deep philosophical divide between frequentist and Bayesian testing, namely that, regardless of the prior distribution used, high-N data that show a significant p-value may at the same time indicate strong evidence in favor of the null hypothesis (Lindley, 1957). This “paradox” is due to Dennis Lindley, one of the most brilliant and influential scholars in statistics.^{1}

Lindley was thoroughly and irrevocably a Bayesian, never passing on the opportunity of being polemic. For example, he argued that “the only good statistics is Bayesian statistics” (Lindley, 1975) or that Bradley Efron, who just received a big price, may have been “falling over all those bootstraps lying around” (Lindley, 1986). He also trashed Taleb’s Black Swan in great style. Somewhat surprisingly, he also took issues with the Bayes factor.^{2}

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.

#### References

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.

#### Footnotes

^{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.