Differential privacy is a privacy measure based on the difficulty of
discriminating between similar input data. In differential privacy analysis,
similar data usually implies that their distance does not exceed a
predetermined threshold. It, consequently, does not take into account the
difficulty of distinguishing data sets that are far apart, which often contain
highly private information. This problem has been pointed out in the research
on differential privacy for static data, and Bayesian differential privacy has
been proposed, which provides a privacy protection level even for outlier data
by utilizing the prior distribution of the data. In this study, we introduce
this Bayesian differential privacy to dynamical systems, and provide privacy
guarantees for distant input data pairs and reveal its fundamental property.
For example, we design a mechanism that satisfies the desired level of privacy
protection, which characterizes the trade-off between privacy and information
utility.

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