  A privacy-constrained information extraction problem is considered where for
a pair of correlated discrete random variables $(X,Y)$ governed by a given
joint distribution, an agent observes $Y$ and wants to convey to a potentially
public user as much information about $Y$ as possible without compromising the
amount of information revealed about $X$. To this end, the so-called {\em
rate-privacy function} is introduced to quantify the maximal amount of
information (measured in terms of mutual information) that can be extracted
from $Y$ under a privacy constraint between $X$ and the extracted information,
where privacy is measured using either mutual information or maximal
correlation. Properties of the rate-privacy function are analyzed and
information-theoretic and estimation-theoretic interpretations of it are
presented for both the mutual information and maximal correlation privacy
measures. It is also shown that the rate-privacy function admits a closed-form
expression for a large family of joint distributions of $(X,Y)$. Finally, the
rate-privacy function under the mutual information privacy measure is
considered for the case where $(X,Y)$ has a joint probability density function
by studying the problem where the extracted information is a uniform
quantization of $Y$ corrupted by additive Gaussian noise. The asymptotic
behavior of the rate-privacy function is studied as the quantization resolution
grows without bound and it is observed that not all of the properties of the
rate-privacy function carry over from the discrete to the continuous case.
