Learned Bayesian Cram\'er-Rao Bound for Unknown Measurement Models Using Score Neural Networks

TL;DR

This paper introduces a fully learned Bayesian Cramér-Rao bound (LBCRB) using score neural networks, enabling estimation bounds without full knowledge of prior and measurement distributions, and demonstrates its effectiveness in various signal processing tasks.

Contribution

The paper proposes two novel approaches to compute the LBCRB by learning prior and measurement distributions with physics-encoded neural networks, incorporating domain knowledge for improved performance.

Findings

The LBCRB approaches accurately estimate bounds in linear and nonlinear signal processing problems.

Incorporating domain knowledge reduces sample complexity and enhances interpretability.

Numerical validation confirms the effectiveness of the proposed methods across multiple applications.

Abstract

The Bayesian Cram\'er-Rao bound (BCRB) is a crucial tool in signal processing for assessing the fundamental limitations of any estimation problem as well as benchmarking within a Bayesian frameworks. However, the BCRB cannot be computed without full knowledge of the prior and the measurement distributions. In this work, we propose a fully learned Bayesian Cram\'er-Rao bound (LBCRB) that learns both the prior and the measurement distributions. Specifically, we suggest two approaches to obtain the LBCRB: the Posterior Approach and the Measurement-Prior Approach. The Posterior Approach provides a simple method to obtain the LBCRB, whereas the Measurement-Prior Approach enables us to incorporate domain knowledge to improve the sample complexity and {interpretability}. To achieve this, we introduce a Physics-encoded score neural network which enables us to easily incorporate such domain knowledge into a neural network. We {study the learning} errors of the two suggested approaches theoretically, and validate them numerically. We demonstrate the two approaches on several signal processing examples, including a linear measurement problem with unknown mixing and Gaussian noise covariance matrices, frequency estimation, and quantized measurement. In addition, we test our approach on a nonlinear signal processing problem of frequency estimation with real-world underwater ambient noise.