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thething: Reviewed pf:thething-prv

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Manos Katsomallos 2021-09-19 23:39:57 +02:00
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text/problem/thething/problem.tex
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@ -173,11 +173,11 @@ Theorem~\ref{theor:thething-prv} proposes how to achieve the desired privacy for
% Due to space constraints, we omit the proof of Theorem~\ref{theor:thething-prv} and defer it for a longer version of this paper. % Due to space constraints, we omit the proof of Theorem~\ref{theor:thething-prv} and defer it for a longer version of this paper.
\begin{proof} \begin{proof}
\label{pf:thething-prv} \label{pf:thething-prv}
All mechanisms use independent randomness, and therefore for a series of events $S_T = {D_1, \dots, D_T}$ and outputs $(\pmb{o}_1, \dots, \pmb{o}_T) \in O \subseteq \mathcal{O}$ it holds that All mechanisms use independent randomness, and therefore for a time series $S_T = {D_1, \dots, D_T}$ and outputs $(\pmb{o}_1, \dots, \pmb{o}_T) \in O \subseteq \mathcal{O}$ it holds that
$$Pr[\mathcal{M}(S_T) = (\pmb{o}_1, \dots, \pmb{o}_T)] = \prod_{i \in [1, T]} Pr[\mathcal{M}_i(D_i) = \pmb{o}_i]$$ $$Pr[\mathcal{M}(S_T) = (\pmb{o}_1, \dots, \pmb{o}_T)] = \prod_{i \in [1, T]} Pr[\mathcal{M}_i(D_i) = \pmb{o}_i]$$
Likewise, for any {\thething}-neighboring series of events $S'_T$ of $S_T$ with the same outputs $(\pmb{o}_1, \dots, \pmb{o}_T) \in O \subseteq \mathcal{O}$ Likewise, for any {\thething}-neighboring time series $S'_T$ of $S_T$ with the same outputs $(\pmb{o}_1, \dots, \pmb{o}_T) \in O \subseteq \mathcal{O}$
$$Pr[\mathcal{M}(S'_T) = (\pmb{o}_1, \dots, \pmb{o}_T)] = \prod_{i \in [1, T]} Pr[\mathcal{M}_i(D'_i) = \pmb{o}_i]$$ $$Pr[\mathcal{M}(S'_T) = (\pmb{o}_1, \dots, \pmb{o}_T)] = \prod_{i \in [1, T]} Pr[\mathcal{M}_i(D'_i) = \pmb{o}_i]$$