  In the application of Bayesian methods to metrology, pre-data probabilities
play a critical role in the estimation of the model uncertainty. Following the
observation that distributions form Riemann's manifolds, methods of
differential geometry can be applied to ensure covariant priors and
uncertainties independent of parameterization. Paradoxes were found in
multi-parameter problems and alternatives were developed; but, when different
parameters are of interest, covariance may be lost. This paper overviews
information geometry, investigates some key paradoxes, and proposes solutions
that preserve covariance.
