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 LDP 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. The relationship between LDP and Maximum Bayesian Privacy (MBP) is first revealed, demonstrating that under uniform prior distribution, a mechanism satisfying $\xi$-LDP will satisfy $\xi$-MBP and conversely $\xi$-MBP also confers 2$\xi$-LDP. Our next theoretical contribution are anchored in the rigorous definitions and relationships between Average Bayesian Privacy (ABP) and Maximum Bayesian Privacy (MBP), encapsulated by equations $\epsilon_{p,a} \leq \frac{1}{\sqrt{2}}\sqrt{(\epsilon_{p,m} + \epsilon)\cdot(e^{\epsilon_{p,m} + \epsilon} - 1)}$. These relationships fortify our understanding of the privacy guarantees provided by various mechanisms. Our work not only lays the groundwork for future empirical exploration but also promises to facilitate the design of privacy-preserving algorithms, thereby fostering the development of trustworthy machine learning solutions.