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

## TL;DR

This paper derives error bounds and provides guidelines for calibrating privacy levels in differentially private Kalman filtering, ensuring accurate state estimation while protecting sensitive data in control systems.

## Contribution

It introduces error and entropy bounds for differentially private Kalman filtering and offers practical guidelines for privacy calibration to maintain desired accuracy.

## Key findings

- Derived bounds on a priori and a posteriori errors.
- Provided entropy bounds for privatized trajectories.
- Demonstrated calibration guidelines through simulations.

## Abstract

Differential privacy has emerged as a formal framework for protecting sensitive information in control systems. One key feature is that it is immune to post-processing, which means that arbitrary post-hoc computations can be performed on privatized data without weakening differential privacy. It is therefore common to filter private data streams. To characterize this setup, in this paper we present error and entropy bounds for Kalman filtering differentially private state trajectories. We consider systems in which an output trajectory is privatized in order to protect the state trajectory that produced it. We provide bounds on a priori and a posteriori error and differential entropy of a Kalman filter which is processing the privatized output trajectories. Using the error bounds we develop, we then provide guidelines to calibrate privacy levels in order to keep filter error within pre-specified bounds. Simulation results are presented to demonstrate these developments.