ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Singular prior and predictive covariances in Gaussian-process regression

Example

On $T = {a, b}$ take mean zero and constant kernel $k (s, t) = 1$ . Observe both indices with independent noise covariance $I_{2}$ , and test the repeated list $X_{*} = (b, b)$ . At observation vector $y = (1, 0)$ , the selected posterior-kernel mean is $(1 / 3, 1 / 3)$ and covariance is $\frac{1}{3} (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}),$ which is singular, although the observed covariance is positive definite.

Facts & Assumptions

Given: AC; the two-index constant-kernel process; and noise $ϵ \sim N_{2} (0, I_{2})$ independent of the process.

[A1]

AC is required by the finite Gaussian-process and multivariate-normal constructions. The Axiom of Choice

[F1]

Any symmetric positive-semidefinite kernel on an arbitrary set defines a canonical cylinder-space Gaussian process under AC. A mean and positive-semidefinite kernel define a Gaussian process

[F2]

Under AC, every finite symmetric positive-semidefinite covariance matrix defines a possibly singular Gaussian law. Multivariate normal law, including singular covariance

[F3]

A finite product of sigma-finite measures exists and has the rectangle product formula. For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique

[F4]

Product-coordinate random elements are independent when their rectangle probabilities factor. Independent random elements Independent random elements are characterized by finite rectangle probabilities

[F5]

Coordinates on the cylinder sigma-algebra are measurable, and a finite product sigma-algebra is generated by its coordinate sets. Coordinate maps, finite-coordinate cylinders, and the cylinder $σ$ -algebra The product sigma-algebra and its finite iterates

[F6]

With positive-definite observed covariance, the finite posterior law is Gaussian with the Schur-complement covariance, which may be singular. Finite Gaussian-process posterior and predictive laws

Verification

Given: The parameters, repeated test list, and observation vector in the Example.

1.1A1F1F2F3F4F5givenconstructalgebra

For any finite coefficient list, the constant kernel's quadratic form is $(\sum_{i} a_{i})^{2} \geq 0$ . Thus [F1] constructs the cylinder GP and [F2] supplies $N_{2} (0, I_{2})$ . Their product probability exists by [F3]; its process and noise random elements are measurable by [F5] and independent by [F4].

1.2F6givenalgebra

The observation covariance is $K_{X X} = J_{2}$ , which has rank one, while $C = J_{2} + I_{2} = (\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array})$ has quadratic form $u^{2} + v^{2} + (u + v)^{2} > 0$ for every nonzero $(u, v)$ . Hence it is positive definite and $C^{- 1} = \frac{1}{3} (\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array})$ . For the repeated test list, $K_{* X} = K_{* *} = J_{2}$ . Since $C$ is positive definite, the hypotheses of [F6] hold.

2.1F6step 1.2algebra

Multiplication gives $K_{* X} C^{- 1} y = (1 / 3, 1 / 3)^{T}$ . Also $C^{- 1} J_{2} = \frac{1}{3} J_{2}$ , so $K_{* X} C^{- 1} K_{X *} = \frac{2}{3} J_{2}$ and the conditional covariance is $J_{2} - \frac{2}{3} J_{2} = \frac{1}{3} J_{2}$ . Its determinant is zero and its rank is one. By [F6] this is a valid singular Gaussian conditional law.

3.1F6step 1.2step 2.1∎

The prior observation covariance and conditional predictive covariance are both singular, whereas the matrix inverted by the formula is the positive-definite observed covariance $C$ . This verifies why singular prior and predictive blocks are permitted under the stated condition.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

50 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

A. W. van der Vaart and J. H. van Zanten, Reproducing kernel Hilbert spaces of Gaussian priors (2008) (standard reference, not scraped)
Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning (MIT Press, 2006), Chapter 2 (standard reference, not scraped)