  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.
