  We study an information theoretic privacy mechanism design problem for two
scenarios where the private data is either observable or hidden. In each
scenario, we first consider bounded mutual information as privacy leakage
criterion, then we use two different per-letter privacy constraints. In the
first scenario, an agent observes useful data $Y$ that is correlated with
private data $X$, and wishes to disclose the useful information to a user. A
privacy mechanism is designed to generate disclosed data $U$ which maximizes
the revealed information about $Y$ while satisfying a bounded privacy leakage
constraint. In the second scenario, the agent has additionally access to the
private data. To this end, we first extend the Functional Representation Lemma
and Strong Functional Representation Lemma by relaxing the independence
condition and thereby allowing a certain leakage to find lower bounds for the
second scenario with different privacy leakage constraints. Furthermore, upper
and lower bounds are derived in the first scenario considering different
privacy constraints. In particular, for the case where no leakage is allowed,
our upper and lower bounds improve previous bounds. Moreover, considering
bounded mutual information as privacy constraint we show that if the common
information and mutual information between $X$ and $Y$ are equal, then the
attained upper bound in the second scenario is tight. Finally, the
privacy-utility trade-off with prioritized private data is studied where part
of $X$, i.e., $X_1$, is more private than the remaining part, i.e., $X_2$, and
we provide lower and upper bounds.
