TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Differential privacy is preserved under randomized postprocessing

Statement

Let $ε \geq 0$ and $0 \leq δ \leq 1$ . Let $K$ be an $(ε, δ)$ -differentially private probability kernel from databases to a measurable space $Y$ , and let $L$ be a data-independent probability kernel from $Y$ to a measurable space $Z$ . Then the composed kernel $K L$ is $(ε, δ)$ -differentially private.

Statement references

The event inequality for probability-kernel mechanisms

Dwork and Roth, The Algorithmic Foundations of Differential Privacy, Proposition 2.1, printed p. 19.

Vadhan, The Complexity of Differential Privacy, §2.1, Lemma 2.1.

Facts & Assumptions

Given: An $(ε, δ)$ -differentially private probability kernel $K$ and a data-independent probability kernel $L$ .

[F1]

Differential privacy means $K (D, A) \leq e^{ε} K (D^{'}, A) + δ$ for every ordered neighboring pair and every measurable event.

[F2]

Probability kernels have measurable event evaluations with probability-measure sections.

[F3]

Bounded-function extension: the one-sided event inequality for probability measures extends to every measurable $g : Y \to [0, 1]$ with the same parameters.

[F4]

Kernel composition is $(K L) (D, B) = \int_{Y} L (y, B) K (D, d y)$ .

[F5]

Kernel composition theorem ensures the displayed composition is a probability kernel.

Proof

1.1F1F2givenchoose

Fix ordered neighboring databases $D \sim D^{'}$ and an event $B$ in the output sigma-algebra of $Z$ . The function $g (y) = L (y, B)$ is measurable and takes values in $[0, 1]$ by [F2].

2.1F1F3F4F5step 1.1algebra∎

Apply [F3] to the event inequality for $K (D, \cdot)$ and $K (D^{'}, \cdot)$ with this $g$ . Then [F4] identifies the two integrals with $(K L) (D, B)$ and $(K L) (D^{'}, B)$ , giving $(K L) (D, B) \leq e^{ε} (K L) (D^{'}, B) + δ$ . The pair and event were arbitrary, and [F5] makes $K L$ a probability kernel, so [F1] proves the claim.

Depends on

Used by

Thresholding a private noisy count Example

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cynthia Dwork and Aaron Roth, The Algorithmic Foundations of Differential Privacy (standard reference, not scraped)
Salil Vadhan, The Complexity of Differential Privacy, 2017 manuscript (standard reference, not scraped)