Information Geometric Approach to Bayesian Lower Error Bounds

TL;DR

This paper introduces a Riemannian metric within information geometry to unify Bayesian and deterministic lower error bounds, including the Barankin bound, improving accuracy in low SNR scenarios.

Contribution

It proposes a new Riemannian metric that simultaneously derives Bayesian and deterministic CRLBs and extends to the Barankin bound, addressing limitations in low SNR conditions.

Findings

Unified framework for Bayesian and deterministic bounds

Extension to Barankin bound for low SNR scenarios

Improved accuracy of error bounds in noisy environments

Abstract

Information geometry describes a framework where probability densities can be viewed as differential geometry structures. This approach has shown that the geometry in the space of probability distributions that are parameterized by their covariance matrix is linked to the fundamentals concepts of estimation theory. In particular, prior work proposes a Riemannian metric - the distance between the parameterized probability distributions - that is equivalent to the Fisher Information Matrix, and helpful in obtaining the deterministic Cram\'{e}r-Rao lower bound (CRLB). Recent work in this framework has led to establishing links with several practical applications. However, classical CRLB is useful only for unbiased estimators and inaccurately predicts the mean square error in low signal-to-noise (SNR) scenarios. In this paper, we propose a general Riemannian metric that, at once, is used to obtain both Bayesian CRLB and deterministic CRLB along with their vector parameter extensions. We also extend our results to the Barankin bound, thereby enhancing their applicability to low SNR situations.