LemmaStatement: AI-adaptedProof: AI-generatedPipeline-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.

A smooth independent residual gives a pointwise density factorization

Statement

Let $C$ be a finite-dimensional real random vector, let $(Y, C)$ have a strictly positive $C^{3}$ joint density $p (y, c)$ , and let $m : R^{d} \to R$ be $C^{3}$ . If $R = Y - m (C)$ is independent of $C$ , then there are strictly positive $C^{3}$ Lebesgue density versions $q_{R}$ and $p_{C}$ such that $p (y, c) = q_{R} (y - m (c)) p_{C} (c)$ for every $(y, c) \in R \times R^{d}$ .

Statement references

Independence of the residual and predictors

The independent joint law is the product of its marginals

Sectionwise translations preserve Lebesgue measure

Facts & Assumptions

Given: The random vector, density, function and independence assumption in the Statement.

[F1]

Translation preserves Lebesgue measure and Lebesgue measurability. Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation

[F2]

Independent random elements have product joint law. Independent random elements have product joint law

[F3]

Nonnegative product-measurable functions can be integrated by iterated sections in any finite coordinate order. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

Proof

Given: The hypotheses in the Statement.

1.1givenF1F3construct

The coordinate change $(y, c) \mapsto (r, c) = (y - m (c), c)$ is a translation in each fixed $c$ -section. By [F1] and iterated integration [F3], the law of $(R, C)$ has density $h (r, c) = p (r + m (c), c)$ . Since $p$ and $m$ are $C^{3}$ and $p > 0$ , $h$ is a strictly positive $C^{3}$ function.

2.1step 1.1givenF2F3construct

Independence and [F2] make the law with density $h (r, c)$ from step 1.1 the product of the marginal laws of $R$ and $C$ . Tonelli [F3] gives marginal density versions, and therefore $h (r, c) = q_{R} (r) p_{C} (c)$ almost everywhere. Choose $c_{0}$ for which $0 < p_{C} (c_{0}) < \infty$ and this equality holds for almost every $r$ . Then $q_{R} (r) = h (r, c_{0}) / p_{C} (c_{0})$ almost everywhere, so this formula supplies a strictly positive $C^{3}$ version of $q_{R}$ .

3.1step 1.1step 2.1F2F3algebra

By Fubini choose $r_{0}$ with $q_{R} (r_{0}) > 0$ such that $h (r_{0}, c) = q_{R} (r_{0}) p_{C} (c)$ for almost every $c$ . Set $p_{C}^{*} (c) = h (r_{0}, c) / q_{R} (r_{0})$ . This is a strictly positive $C^{3}$ function and equals $p_{C}$ almost everywhere. Thus $h (r, c) = q_{R} (r) p_{C}^{*} (c)$ almost everywhere; both sides are continuous, so the equality holds everywhere.

4.1step 1.1step 3.1algebra∎

Substituting $r = y - m (c)$ into step 3.1 yields $p (y, c) = q_{R} (y - m (c)) p_{C}^{*} (c)$ for every $(y, c)$ . Renaming $p_{C}^{*}$ as $p_{C}$ proves the Statement.

References

Peters et al., JMLR (2014), Appendix A.15, proof of Theorem 34

Depends on

Used by

Population identifiability and RESIT correctness for restricted ANMs Theorem

Dependency tree · two levels

29 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

Jonas Peters, Joris Mooij, Dominik Janzing, and Bernhard Schölkopf, Causal Discovery with Continuous Additive Noise Models, Appendix A.15 (standard reference, not scraped)