  Using differential geometry, I derive a form of the Bayesian Cram\'er-Rao
bound that remains invariant under reparametrization. With the invariant
formulation at hand, I find the optimal and naturally invariant bound among the
Gill-Levit family of bounds. By assuming that the prior probability density is
the square of a wavefunction, I also express the bounds in terms of functionals
that are quadratic with respect to the wavefunction and its gradient. The
problem of finding an unfavorable prior to tighten the bound for minimax
estimation is shown, in a special case, to be equivalent to finding the ground
state of a Schr\"odinger equation, with the Fisher information playing the role
of the potential. To illustrate the theory, two quantum estimation problems,
namely, optomechanical waveform estimation and subdiffraction incoherent
optical imaging, are discussed.
