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Differential privacy has emerged as a formal framework for protectingsensitive information in control systems. One key feature is that it is immuneto post-processing, which means that arbitrary post-hoc computations can beperformed on privatized data without weakening differential privacy. It istherefore common to filter private data streams. To characterize this setup, inthis paper we present error and entropy bounds for Kalman filteringdifferentially private state trajectories. We consider systems in which anoutput trajectory is privatized in order to protect the state trajectory thatproduced it. We provide bounds on a priori and a posteriori error anddifferential entropy of a Kalman filter which is processing the privatizedoutput trajectories. Using the error bounds we develop, we then provideguidelines to calibrate privacy levels in order to keep filter error withinpre-specified bounds. Simulation results are presented to demonstrate thesedevelopments.

Error Bounds and Guidelines for Privacy Calibration in Differentially Private Kalman Filtering