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

2025-02-02

AI Summary

In Bayesian estimation, the Bayesian Cramér–Rao bound (BCRB) is intractable when both the prior and measurement model are unknown and nonlinear, precluding analytical derivation. Method: This paper proposes a learnable BCRB (LBCRB) framework—introducing two data-driven learning paradigms: posterior-based and measurement-prior-based learning. A physics-informed score neural network is designed, embedding domain knowledge to enhance interpretability and sample efficiency. Contributions/Results: LBCRB enables the first model-free estimation of the BCRB under unknown nonlinear measurement models. Theoretical analysis establishes a learning error bound. Experiments validate its effectiveness across diverse scenarios: linear systems with unknown mixing matrices, frequency estimation, quantized measurements, and real-world nonlinear underwater acoustic noise. In all cases, LBCRB tightly approximates the true BCRB, demonstrating superior accuracy and generalizability.

Technology Category

• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Signal Processing
• Automatic Learning
• Bayesian Cramer-Rao Bound

Innovation

• LBCRB method
• Neural Networks
• Physical Encoding Grading Neural Network