DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Gaussian partial correlation

Definition

Assume AC. Let $X = (X_{1}, \dots, X_{p})$ have a nondegenerate multivariate-normal law with mean $μ$ and positive-definite covariance matrix $Σ$ . For distinct indices $i, j$ and $S \subseteq {1, \dots, p} ∖ {i, j}$ , define the conditional covariance block

K_{i j ∣ S} = Σ_{{i, j}, {i, j}} - Σ_{{i, j}, S} Σ_{S, S}^{- 1} Σ_{S, {i, j}},

with $K_{i j ∣ \emptyset} = Σ_{{i, j}, {i, j}}$ . Whenever the diagonal entries of this block are positive, the partial correlation is

ρ_{i j ∣ S} = \frac{(K_{i j ∣ S})_{12}}{\sqrt{(K_{i j ∣ S})_{11} (K_{i j ∣ S})_{22}}} .

The nondegenerate Gaussian case makes the conditioning covariance block invertible; the following theorem proves that $K_{i j ∣ S}$ is positive definite, so this definition applies to every such $i, j, S$ .

Definition references

Multivariate normal law, including singular covariance

The Axiom of Choice

Depends on

Used by

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

Kalisch and Bühlmann, Estimating High-Dimensional Directed Acyclic Graphs with the PC-Algorithm, §2.2.2 (standard reference, not scraped)
Lauritzen, Graphical Models lecture notes, §4.1.3, Proposition 4.2, pp. 97–98 (standard reference, not scraped)