  This paper considers reparameterization invariant Bayesian point estimates
and credible regions of model parameters for scientific inference and
communication. The effect of intrinsic loss function choice in Bayesian
intrinsic estimates and regions is studied with the following findings. A
particular intrinsic loss function, using Kullback-Leibler divergence from the
full model to the restricted model, has strong connection to a Bayesian
predictive criterion, which produces point estimates with the best predictive
performance. An alternative intrinsic loss function, using Kullback-Leibler
divergence from the restricted model to the full model, produces estimates with
interesting frequency properties for at least some commonly used distributions,
that is, unbiased minimum variance estimates of the location and scale
parameters.
