Physics-inspired forms of the Bayesian Cram\'er-Rao bound

TL;DR

This paper introduces a geometry-based, invariant form of the Bayesian Cramér-Rao bound, linking it to quantum mechanics and demonstrating its application in quantum estimation problems like optomechanical waveform estimation and optical imaging.

Contribution

It derives a reparametrization-invariant Bayesian Cramér-Rao bound using differential geometry and connects the minimax estimation problem to the Schrödinger equation.

Findings

Derived an invariant Bayesian Cramér-Rao bound using differential geometry.

Linked the minimax estimation problem to the Schrödinger equation.

Applied the theory to quantum estimation problems in optics.

Abstract

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.