Abstract
In a recent article in Science on "Bayes' Theorem in the 21st Century", Bradley Efron uses Bayes' theorem to calculate the probability that twins are identical given that the sonogram shows twin boys. He concludes that Bayesian calculations cannot be uncritically accepted when using uninformative priors. We argue that this conclusion is problematic because Efron's example on identical twins does not use data, hence it is not Bayesian statistics; his priors are not appropriate and are not uninformative; and using the available data point and an uninformative prior actually leads to a reasonable posterior distribution.
Highlights
We have other more technical issues with Efron’s example
While we wholeheartedly agree that statistical results should not be uncritically accepted, we find Efron’s example ineffective in showing that Bayesian statistics require more careful checking than any other kind of statistics
For the uninformative prior, Efron mentions erroneously that he used a uniform distribution between zero and one, which is clearly different from the value of 0.5 that was used
Summary
We have other more technical issues with Efron’s example. Efron interprets the term P(A) on the right side of the equation (see sidebar in Efron 2013a1) as the prior on the probability that twins are identical. In his example on uninformative priors, Efron uses Bayes’ theorem to calculate the probability that twins are identical given that the sonogram shows twin boys. The probability of identical twins given the twins are two boys: 1) Uninformative prior: p(θ|x) = Beta(1+x,1+n-x) = Beta(1+0,1+1-0) = Beta(1, 2), which has an expected value of 0.33 and a 95% interval of 0.013 – 0.84.
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