  We investigate the problem of estimating a random variable $Y\in \mathcal{Y}$
under a privacy constraint dictated by another random variable $X\in
\mathcal{X}$, where estimation efficiency and privacy are assessed in terms of
two different loss functions. In the discrete case, we use the Hamming loss
function and express the corresponding utility-privacy tradeoff in terms of the
privacy-constrained guessing probability $h(P_{XY}, \epsilon)$, the maximum
probability $\mathsf{P}_\mathsf{c}(Y|Z)$ of correctly guessing $Y$ given an
auxiliary random variable $Z\in \mathcal{Z}$, where the maximization is taken
over all $P_{Z|Y}$ ensuring that $\mathsf{P}_\mathsf{c}(X|Z)\leq \epsilon$ for
a given privacy threshold $\epsilon \geq 0$. We prove that $h(P_{XY}, \cdot)$
is concave and piecewise linear, which allows us to derive its expression in
closed form for any $\epsilon$ when $X$ and $Y$ are binary. In the non-binary
case, we derive $h(P_{XY}, \epsilon)$ in the high utility regime (i.e., for
sufficiently large values of $\epsilon$) under the assumption that $Z$ takes
values in $\mathcal{Y}$. We also analyze the privacy-constrained guessing
probability for two binary vector scenarios. When $X$ and $Y$ are continuous
random variables, we use the squared-error loss function and express the
corresponding utility-privacy tradeoff in terms of $\mathsf{sENSR}(P_{XY},
\epsilon)$, which is the smallest normalized minimum mean squared-error (mmse)
incurred in estimating $Y$ from its Gaussian perturbation $Z$, such that the
mmse of $f(X)$ given $Z$ is within $\epsilon$ of the variance of $f(X)$ for any
non-constant real-valued function $f$. We derive tight upper and lower bounds
for $\mathsf{sENSR}$ when $Y$ is Gaussian. We also obtain a tight lower bound
for $\mathsf{sENSR}(P_{XY}, \epsilon)$ for general absolutely continuous random
variables when $\epsilon$ is sufficiently small.
