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

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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.
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A Gaussian process retains uncertainty at an independent index

Example

Let $T = {a, b}$ , let $m (a) = m (b) = 0$ , and set $k (s, t) = 1$ if $s = t$ and $0$ otherwise. Observe only $f_{a}$ with independent noise variance one and predict $f_{b}$ . The posterior-kernel law of $f_{b}$ is $N (0, 1)$ for every observation value, so observing the independent coordinate does not reduce uncertainty at $b$ .

Facts & Assumptions

Given: AC; the two-index equality kernel; and an independent $N (0, 1)$ observation-noise variable.

[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 stated mean and covariance. Finite Gaussian-process posterior and predictive laws

Verification

Given: The two-index model and one noisy observation in the Example.

1.1A1F1F2F3F4F5givenconstructalgebra

For any finite list of indices, group coefficients at repeated sites. The equality kernel's quadratic form is the sum, over distinct sites $t$ , of $(\sum_{i : t_{i} = t} a_{i})^{2}$ , hence is nonnegative. By [F1] the canonical GP law exists and [F2] supplies the $N (0, 1)$ noise law. Their product probability exists by [F3]; its process and noise random elements are measurable by [F5] and independent by [F4].

1.2givenF6algebra

Here $K_{X X} = 1$ , $K_{* *} = 1$ , $K_{* X} = K_{X *} = 0$ , and the noise covariance is $R = 1$ . Thus $C = K_{X X} + R = 2 > 0$ , so [F6] applies.

2.1F6step 1.2algebra∎

The formula in [F6] gives posterior mean $0 + 0 \cdot 2^{- 1} (y - 0) = 0$ and variance $1 - 0 \cdot 2^{- 1} \cdot 0 = 1$ for every $y \in R$ . Hence the posterior version at the unobserved independent index is its original $N (0, 1)$ law.

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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)