CorollaryStatement: AI-adaptedProof: 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.

Finite Gaussian-process regression posterior and predictive law

Statement

Assume AC. Let $X$ be a set, $m : X \to R$ , and $k : X \times X \to R$ a symmetric kernel whose every finite Gram quadratic form on points of $X$ is nonnegative. Let $f = (f_{t})_{t \in X}$ be the canonical Gaussian process with $m (t) = E [f_{t}]$ and $k (s, t) = E [(f_{s} - m (s)) (f_{t} - m (t))]$ . For finite ordered lists $A = (a_{i})_{i = 1}^{r}$ and $B = (b_{j})_{j = 1}^{s}$ in $X$ , set $K_{A B} = (k (a_{i}, b_{j}))$ ; repetitions are allowed. Take $X = (x_{i})_{i = 1}^{n} \in X^{n}$ and $X_{*} = (x_{j}^{*})_{j = 1}^{p} \in X^{p}$ with $n, p \geq 1$ , and write $m_{X} = (m (x_{i}))$ , $m_{*} = (m (x_{j}^{*}))$ , $f_{X} = (f_{x_{i}})$ , and $f_{*} = (f_{x_{j}^{*}})$ . Let $ε \sim N_{n} (0, R)$ be independent of $f$ , where $R$ is symmetric positive semidefinite, and set $Y = f_{X} + ε$ and $A_{X} = K_{X X} + R$ . Suppose $A_{X}$ is positive definite. Then for every $y \in R^{n}$ the conditional law of $f_{*}$ given $Y = y$ has mean $m_{*} + K_{* X} A_{X}^{- 1} (y - m_{X})$ and covariance $K_{* *} - K_{* X} A_{X}^{- 1} K_{X *} .$ This covariance is positive semidefinite and may be singular. If $η_{*} \sim N_{p} (0, R_{*})$ is independent of the pair $(f, ε)$ , where $R_{*}$ is symmetric positive semidefinite, then the conditional law of $f_{*} + η_{*}$ has the same mean and covariance with $R_{*}$ added. The displayed kernel is defined for every $y$ , although regular conditional versions are determined only $P_{Y}$ -almost surely.

Statement references

Rasmussen and Williams, GPML, Chapter 2 §2.2, equations (2.18)–(2.24), printed pp. 15–18

Bishop, PRML, §6.4.2, equations (6.61)–(6.68), printed pp. 306–309

Facts & Assumptions

Given: AC; a set $X$ , mean $m$ , symmetric positive-semidefinite kernel $k$ ; finite ordered lists $X$ and $X_{*}$ of positive lengths; observation noise $ε \sim N_{n} (0, R)$ independent of the process with $R ⪰ 0$ ; optional test noise $η_{*} \sim N_{p} (0, R_{*})$ independent of the pair $(f, ε)$ with $R_{*} ⪰ 0$ ; and $A_{X} = K_{X X} + R ≻ 0$ .

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. The Axiom of Choice

[F1]

Under AC, a mean function and symmetric positive-semidefinite kernel on an arbitrary index set determine a unique canonical process on the cylinder sigma-algebra with the stated finite Gaussian marginals; the construction supplies no sample-path continuity. A mean and positive-semidefinite kernel define a Gaussian process

[F2]

A real Gaussian process has multivariate-normal finite evaluation vectors, including repeated indices and singular covariance matrices. Gaussian process

[F3]

For finite lists, independent Gaussian observation noise with PSD covariance and positive-definite observed covariance gives the stated all- $y$ conditional Gaussian kernel; its predictive covariance may be singular, and Gaussian test noise independent of the pair $(f, ε)$ adds its covariance. Finite Gaussian-process posterior and predictive laws

[F4]

In this model, $A_{X} = K_{X X} + R$ is invertible exactly when it is positive definite. Positive definiteness is exactly the inverse condition for the observed GP covariance

Proof

Given: The assumptions in the Statement.

1.1A1F1F2givenconstruct

By [A1] and [F1], the specified mean and kernel give the canonical cylinder-space Gaussian process. By [F2], its finite evaluations on $X$ and $X_{*}$ are jointly multivariate normal, including when either list has repetitions or a covariance block is singular.

2.1A1F3F4givenstep 1.1∎

In the notation of the finite model, the observed block in [F3] is $C = K_{X X} + R = A_{X}$ . By the hypothesis and [F4], $C ≻ 0$ , so every inverse required by [F3] exists. Under the optional noisy-prediction hypothesis, $η_{*}$ is independent of the pair $(f, ε)$ , exactly the joint-independence premise for [F3]'s Gaussian stack. Applying [F3] with these lists, means, noise covariance, and observed block gives the displayed conditional mean and covariance; its statement also gives positive semidefiniteness, possible singularity, the all- $y$ version, and the test-noise covariance addition.

Depends on

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Sources

Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 2 §2.2 (standard reference, not scraped)
Christopher M. Bishop, Pattern Recognition and Machine Learning, §6.4.2 (standard reference, not scraped)