I came across the notion of information theoretic privacy in the paper of Yamamoto ("A source coding problem for sources with additional outputs to keep secret from the receiver or wiretappers "). The way this problem is formulated is as follows.

We have a random variable $X$ and a correlated random variable $Y$ with joint distribution $q(x,y)$. We want to keep $X$ as private as possible while sending $Y$ withing distortion $D$ with the least compression rate (like conventional rate distortion problem). The fundamental limit for the rate is $$R(D,L):=\inf_{p_{\hat{Y}|Y}\in\mathcal{P}(D,L)} I(Y;\hat{Y})$$ where $\mathcal{P}(D,L)$ is the collection of all joint distribution $p(x,y,\hat{y})$ such that $\sum_{\hat{y}}p(x,y,\hat{y})=q(x,y)$, $Ed(Y,\hat{Y})\leq D$ and $I(X; \hat{Y})\leq L$.

Now I want to look at this problem from different angle. Random variable $Y$ is generated from $X$ and $Z$ is generated from $Y$. The quantity to look at now is $$\sup \{D_1(P_X||P_Y), \text{while}~D_2(P_X||P_Z) \geq \gamma\}$$ where $D_1$ and $D_2$ are two different probability metrics.

My questions:

- Do you think it is a wise approach to modify the privacy problem?
- How to choose the probability metrics $D_1$ and $D_2$ as to make the first problem as special case of the second one?

Any idea is appreciated,