Finite-source privacy–utility region

Let $S$ be a finite sensitive variable and $X$ a finite useful variable with joint law $P_{SX}$. For each finite nonempty release alphabet $\mathcal{Z}$, choose a stochastic kernel $K(z\mid x)$ from $\mathcal{X}$ to $\mathcal{Z}$ (Finite stochastic kernels) and form

$P_{SXZ}(s,x,z):=P_{SX}(s,x)K(z\mid x)\mathrm{.}$

Thus the induced variables satisfy the explicitly defined factorization $P_{SXZ}=P_{SX}{K}_{Z\mid X}$, written $S-X-Z$. The privacy–utility region for this source is the closure in ${\mathbb{R}}^2$ of

${\bigcup }_{\mathcal{Z},K}\left\{\right(I(S;Z),I(X;Z)\left)\right\},$

where mutual information uses nats and the finite definition Finite mutual information. The first coordinate is the chosen leakage metric and the second is the chosen retained-information utility. This model does not itself impose a distortion, rate, or perfect-privacy constraint; those define different optimization problems.