Can Bayes give second-order reproducibility?

We consider default priors for Bayes analysis as initiated in Bayes (1763), then Laplace (1812), Jeffreys (1961), Bernardo (1979), and many more, and viewed recently as “potentially dangerous” (Efron, 2013) or potentially useful (Fraser, 2013). We use nominal data size n to develop an explicit new prior that has second-order O(n−1) reproducibility for any specified parameter of interest. As part of this we see that untargetted Bayes inference is routinely just first-order O(n−1/2), and thus comparable to maximum likelihood and signed-likelihood-root methods using Central Limit Theorem Normality. A related result is that such mathematical priors do not give valid probability statements by means of the conditional probability lemma, as necessarily they are just mathematical, without physical reference. But they can acquire properties based on repetition or reproducibility as proposed later by Fisher (1930) but implicitly in Laplace (1812). This of course means confidence which in turn was not otherwise developed at the time of Bayes or Laplace.