Inverting the Fisher information operator in non-linear models

TL;DR

This paper demonstrates that in certain non-linear models, the Fisher information operator can be inverted under specific conditions, enabling the derivation of optimal Bayesian algorithms for inverse problems involving PDEs.

Contribution

It establishes conditions for invertibility of the Fisher information operator in non-linear models and constructs the associated efficient Gaussian for optimal Bayesian inference.

Findings

Fisher information operator is invertible when the model's score is injective.

Provides a characterization of the well-identified Hilbert spaces.

Demonstrates the approach on PDE models like reaction-diffusion and Navier-Stokes.

Abstract

We consider non-linear regression models corrupted by generic noise when the regression functions form a non-linear subspace of L^2, relevant in non-linear PDE inverse problems and data assimilation. We show that when the score of the model is injective, the Fisher information operator is automatically invertible between well-identified Hilbert spaces, and we provide an operational characterization of these spaces. This allows us to construct in broad generality the efficient Gaussian involved in the classical minimax and convolution theorems to establish information lower bounds, that are typically achieved by Bayesian algorithms thus showing optimality of these methods. We illustrate our results on time-evolution PDE models for reaction-diffusion and Navier-Stokes equations.