  We examine the relationship between privacy metrics that utilize information
density to measure information leakage between a private and a disclosed random
variable. Firstly, we prove that bounding the information density from above or
below in turn implies a lower or upper bound on the information density,
respectively. Using this result, we establish new relationships between local
information privacy, asymmetric local information privacy, pointwise maximal
leakage and local differential privacy. We further provide applications of
these relations to privacy mechanism design. Furthermore, we provide statements
showing the equivalence between a lower bound on information density and
risk-averse adversaries. More specifically, we prove an equivalence between a
guessing framework and a cost-function framework that result in the desired
lower bound on the information density.
