  The incorporation of priors in the Optimal Uncertainty Quantification (OUQ)
framework \cite{OSSMO:2011} reveals brittleness in Bayesian inference; a model
may share an arbitrarily large number of finite-dimensional marginals with, or
be arbitrarily close (in Prokhorov or total variation metrics) to, the
data-generating distribution and still make the largest possible prediction
error after conditioning on an arbitrarily large number of samples. The initial
purpose of this paper is to unwrap this brittleness mechanism by providing (i)
a quantitative version of the Brittleness Theorem of \cite{BayesOUQ} and (ii) a
detailed and comprehensive analysis of its application to the revealing example
of estimating the mean of a random variable on the unit interval $[0,1]$ using
priors that exactly capture the distribution of an arbitrarily large number of
Hausdorff moments.
  However, in doing so, we discovered that the free parameter associated with
Markov and Kre\u{\i}n's canonical representations of truncated Hausdorff
moments generates reproducing kernel identities corresponding to reproducing
kernel Hilbert spaces of polynomials.
  Furthermore, these reproducing identities lead to biorthogonal systems of
Selberg integral formulas.
  This process of discovery appears to be generic: whereas Karlin and Shapley
used Selberg's integral formula to first compute the volume of the Hausdorff
moment space (the polytope defined by the first $n$ moments of a probability
measure on the interval $[0,1]$), we observe that the computation of that
volume along with higher order moments of the uniform measure on the moment
space, using different finite-dimensional representations of subsets of the
infinite-dimensional set of probability measures on $[0,1]$ representing the
first $n$ moments, leads to families of equalities corresponding to classical
and new Selberg identities.
