  Privacy-preserving data release mechanisms aim to simultaneously minimize
information-leakage with respect to sensitive data and distortion with respect
to useful data. Dependencies between sensitive and useful data results in a
privacy-utility tradeoff that has strong connections to generalized
rate-distortion problems. In this work, we study how the optimal
privacy-utility tradeoff region is affected by constraints on the data that is
directly available as input to the release mechanism. In particular, we
consider the availability of only sensitive data, only useful data, and both
(full data). We show that a general hierarchy holds: the tradeoff region given
only the sensitive data is no larger than the region given only the useful
data, which in turn is clearly no larger than the region given both sensitive
and useful data. In addition, we determine conditions under which the tradeoff
region given only the useful data coincides with that given full data. These
are based on the common information between the sensitive and useful data. We
establish these results for general families of privacy and utility measures
that satisfy certain natural properties required of any reasonable measure of
privacy or utility. We also uncover a new, subtler aspect of the data
processing inequality for general non-symmetric privacy measures and discuss
its operational relevance and implications. Finally, we derive exact
closed-analytic-form expressions for the privacy-utility tradeoffs for
symmetrically dependent sensitive and useful data under mutual information and
Hamming distortion as the respective privacy and utility measures.
