# Refinement Orders for Quantitative Information Flow and Differential Privacy +
## 8 Conclusions  
We have investigated various refinement orders for mechanisms for information protection, combining the max-case perspective typical of DP and its variants with the robustness of the QIF approach. We have provided structural characterizations of these preorders and methods to verify them efficiently. Then, we have considered various DP mechanisms, and investigated the relation between the \(\varepsilon\)-based measurement of privacy and our orders. We have shown that, while within the same family of mechanisms, a smaller \(\varepsilon\) implies the refinement order, this is almost never the case for mechanisms belonging to different families. **Author Contributions:** Conceptualization, K.C., N.F. and C.P. ; methodology, K.C., N.F. and C.P. ; software, K.C. and N.F. ; validation, K.C. and N.F. ; formal analysis, K.C. and N.F. ; investigation, K.C. and N.F. ; resources, C.P. ; data curation, K.C. and N.F. ; writing--original draft preparation, K.C. and N.F. ; writing--review and editing, K.C., N.F. and C.P. ; visualization, K.C. and N.F. ; supervision, K.C. and C.P. ; project administration, K.C. and C.P. ; funding acquisition, C.P. All authors have read and agreed to the published version of the manuscript. **Funding:** This work has been supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, Grant agreement No. 835294, by the project ANR-16-CE25-0011 REPAS, and by the Equipe Associee LOGIS. **Conflicts of Interest:** The authors declare no conflict of interest. **Appendix A. Proofs of Results about the Max-Case Refinement**  
**Theorem A1**. : _Let \(\pi\colon\mathbb{D}\mathcal{X}\). If \(A\subseteq^{\max}B\), then the posteriors of \(B\) (under \(\pi\)) are convex-combinations of those of \(A\), that is,_  
\[supp\left[\pi,B\right]\quad\subseteq\quad ch\,supp\left[\pi,A\right]\] (A1)  
_Moreover, if (A1) holds and \(\pi\) is full support, then \(A\subseteq^{\max}B\)._  
**Proof. ** Note that seeing \(\pi\) as a row vector, \(\pi A\) and \(\pi B\) are the output distributions of \(A\) and \(B\), respectively. Denote by \(\alpha^{y}\) and \(\beta^{z}\) the posteriors of \(\left[\pi,A\right]\) and \(\left[\pi,B\right]\), respectively; we have as many posteriors as the elements in the support of the output distributions, that is, for each \(y\colon supp\,\pi A,z\colon supp\,\pi B\). (A1) can be written as  
\[\forall z\colon supp\,\pi B.\,\left(\beta^{z}=\sum_{y}c_{y}^{z}\alpha^{y}\quad \text{where}\quad\sum_{y}c_{y}^{z}=1\right)\] (A2)  
The proof consists of two parts: first, we show that (A2) for uniform \(\pi\) is _equivalent_ to \(A\subseteq^{\max}B\). Second, we show that (A2) for full-support \(\pi\)_implies_ (A2) for any other prior. For the first part, letting \(\pi\) be uniform, we show that (A2) is equivalant to \(R\tilde{A}=\tilde{B}\). This is easy to see since since the \(y\)-th row of \(\tilde{A}\) is \(\alpha^{y}\) and the \(z\)-th row of \(\tilde{B}\) is \(\beta^{z}\). Hence, we can construct \(R\) from the convex coefficients, and vice versa, as \(R_{z,y}=c_{y}^{z}\). For the second part, let \(\pi:\mathbb{D}\mathcal{X}\) be full-support and \(\hat{\pi}\colon\mathbb{D}\mathcal{X}\) be arbitrary. Since \(supp\,\hat{\pi}\subseteq supp\,\pi\), we necessarily have \(supp\,\hat{\pi}C\subseteq supp\,\hat{\pi}C\) for any channel \(C\). Assume that (A1) holds for \(\pi\) and let \(c_{y}^{z}\) be the corresponding convex coefficients.