Corrections to “Distribution Privacy Under Function Recoverability”

In the article <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> , in the proof of Lemma 2, the sentence after (10) “For a fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q^{(n)}$ </tex-math></inline-formula> , since <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(P_{X}W)^{n}(z^{n})$ </tex-math></inline-formula> is the same for all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z^{n}\in \mathcal {T}_{Q^{(n)}}$ </tex-math></inline-formula> , if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widehat {P}_{n}(z^{n})$ </tex-math></inline-formula> were to vary across <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z^{n}\in \mathcal {T}_{Q^{(n)}}$ </tex-math></inline-formula> , the querier can pick that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tilde {z^{n}}$ </tex-math></inline-formula> , say, in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {T}_{Q^{(n)}}$ </tex-math></inline-formula> for which <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D\left ({P_{X}\big |\big |\widehat {P}_{n}\left ({\tilde {z^{n}}}\right)}\right)$ </tex-math></inline-formula> is smallest over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {T}_{Q^{(n)}}$ </tex-math></inline-formula> and use <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widehat {P}_{n}\left ({\tilde {z^{n}}}\right)$ </tex-math></inline-formula> as the estimate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{X}$ </tex-math></inline-formula> for all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z^{n}\in \mathcal {T}_{Q^{(n)}}$ </tex-math></inline-formula> , denoting it by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widehat {P}_{n}\left ({Q^{(n)}}\right)$ </tex-math></inline-formula> ; this will only serve to decrease the right-side of (10), bearing in mind the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\inf $ </tex-math></inline-formula> with respect to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\widehat {P}_{n}$ </tex-math></inline-formula> in the left-side of (5)” is incorrect.