Lindley's paradox describes a counterintuitive situation in statistics in which the Bayesian and frequentist approaches to a hypothesis testing problem give opposite results for certain choices of the prior distribution. The problem of the disagreement between the two approaches was discussed in Harold Jeffreys' textbook[1]; it became known as Lindley's paradox after Dennis Lindley called the disagreement a paradox in a 1957 paper[2].
Description of the paradox
Consider a null hypothesis H0, the result of an experiment x, and a prior distribution that favors H0 weakly. Lindley's paradox occurs when
- The result x is significant by a frequentist test, indicating sufficient evidence to reject H0, say, at the 5% level, and
- The posterior probability of H0 given x is high, say, 95%, indicating strong evidence that H0 is in fact true.
These results can happen at the same time when the prior distribution is the sum of a sharp peak at H0 with probability p and a broad distribution with the rest of the probability 1 − p. It is a result of the prior having a sharp feature at H0 and no sharp features anywhere else.
Notes
- ^ Jeffreys, Harold (1939). Theory of Probability. Oxford University Press. MR924.
- ^ Lindley, D.V. (1957). "A Statistical Paradox". Biometrika 44 (1-2): 187–192. doi:.
References
- Shafer, Glenn (1982). "Lindley's paradox". Journal of the American Statistical Association 77 (378): 325–334. doi:. MR664677.
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