Identical twins and Bayes' theorem in the 21st century

In an 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. While we agree that the choice of the prior is essential, we argue that the calculations on identical twins give a biased impression of the influence of uninformative priors in Bayesian data analyses.