Title: Deciphering the Interplay between Local Differential Privacy, Average Bayesian Privacy, and Maximum Bayesian Privacy

Abstract: The swift evolution of machine learning has led to emergence of various definitions of privacy due to the threats it poses to privacy, including the concept of local differential privacy (LDP). Although widely embraced and utilized across numerous domains, this conventional approach to measure privacy still exhibits certain limitations, spanning from failure to prevent inferential disclosure to lack of consideration for the adversary's background knowledge. In this comprehensive study, we introduce Bayesian privacy and delve into the intricate relationship between local differential privacy and its Bayesian counterparts, unveiling novel insights into utility-privacy trade-offs. We introduce a framework that encapsulates both attack and defense strategies, highlighting their interplay and effectiveness. Our theoretical contributions are anchored in the rigorous definitions and relationships between Average Bayesian Privacy (ABP) and Maximum Bayesian Privacy (MBP), encapsulated by equations $\epsilon_{p,a} \leq \frac{1}{\sqrt{2}}\sqrt{(\epsilon_{p,m} + \epsilon)\cdot(e^{\epsilon_{p,m} + \epsilon} - 1)}$ and the equivalence between $\xi$-MBP and $2\xi$-LDP established under uniform prior distribution. These relationships fortify our understanding of the privacy guarantees provided by various mechanisms, leading to the realization that a mechanism satisfying $\xi$-LDP also confers $\xi$-MBP, and vice versa. Our work not only lays the groundwork for future empirical exploration but also promises to enhance the design of privacy-preserving algorithms that do not compromise on utility, thereby fostering the development of trustworthy machine learning solutions.