Privacy-Preserving Cram\'er-Rao Lower Bound

Abstract

This paper establishes the privacy-preserving Cram\'er-Rao (CR) lower bound theory, characterizing the fundamental limit of identification accuracy under privacy constraint. An identifiability criterion under privacy constraint is derived by using Fisher information matrix as the privacy metric. In the identifiable case, the privacy-preserving CR lower bound is established and its attainability is demonstrated, thereby ensuring the existence of the privacy-preserving Fisher information matrix with explicit expression. Then, the privacy-preserving CR lower bound theory is extended to the multi-sensor multi-measurement system. Specifically, the additivity principle of privacy-preserving Fisher information matrices across both spatial and temporal dimensions is established, building a relationship between privacy-preserving CR lower bounds for the multi-sensor multi-measurement system and its subsystems. Using this additivity principle, distributed identification algorithms capable of achieving the privacy-preserving CR lower bound are further proposed. Numerical examples are provided to demonstrate the privacy-preserving CR lower bound and show the effectiveness of the proposed algorithms.