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.
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A nonzero-mean Gaussian process with correlated observation noise

Example

Let $T = {a, b, c}$ , let $m (a) = 1$ , $m (b) = - 1$ , $m (c) = 2$ , and let $k (s, t) = 1$ for all $s, t \in T$ . Observe at $X = (a, b)$ with noise covariance $R = (\begin{array}{cc} 1 & 1 / 2 \\ 1 / 2 & 1 \end{array})$ and evaluate the selected Gaussian conditional-kernel version at $y = (0, 1)$ for the test index $c$ . The posterior law is $N (16 / 7, 3 / 7)$ .

Facts & Assumptions

Given: AC; the finite index set and mean above; the constant covariance kernel; and observation noise with law $N_{2} (0, R)$ .

[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 $C$ , the Gaussian conditional kernel has mean $m_{*} + K_{* X} C^{- 1} (y - m_{X})$ and covariance $K_{* *} - K_{* X} C^{- 1} K_{X *}$ . Finite Gaussian-process posterior and predictive laws

Verification

Given: The parameters and observed value in the Example.

1.1A1F1F2F3F4F5givenconstructalgebra

For every finite coefficient list on $T$ , the constant kernel has quadratic form $(\sum_{i} a_{i})^{2} \geq 0$ . The noise matrix is positive definite because $u^{2} + u v + v^{2} = (u + v / 2)^{2} + 3 v^{2} / 4 > 0$ for $(u, v) \neq (0, 0)$ . By [F1] there is a cylinder-space GP law $μ$ and [F2] gives the noise law. Their product probability exists by [F3]; its process and noise coordinates are measurable by [F5] and independent by [F4]. Thus this is a model satisfying the theorem's noise assumption.

1.2givenalgebra

The observed and test covariance blocks are $K_{X X} = J_{2}$ , $K_{c X} = (1, 1)$ , and $K_{c c} = 1$ . Thus $C = J_{2} + R = (\begin{array}{cc} 2 & 3 / 2 \\ 3 / 2 & 2 \end{array})$ , which is positive definite since its quadratic form is $2 (u + 3 v / 4)^{2} + 7 v^{2} / 8$ . Direct multiplication gives $C^{- 1} = (\begin{array}{cc} 8 / 7 & - 6 / 7 \\ - 6 / 7 & 8 / 7 \end{array})$ .

2.1F6step 1.2algebra

Since $m_{X} = (1, - 1)$ , $m_{c} = 2$ , and $y - m_{X} = (- 1, 2)$ , $C^{- 1} (y - m_{X}) = (- 20 / 7, 22 / 7)$ and $K_{c X} C^{- 1} (y - m_{X}) = 2 / 7$ . The posterior mean in [F6] is therefore $2 + 2 / 7 = 16 / 7$ .

3.1F6step 1.2step 2.1algebra∎

Also $C^{- 1} K_{X c} = (2 / 7, 2 / 7)^{T}$ , so $K_{c X} C^{- 1} K_{X c} = 4 / 7$ . The posterior variance is $1 - 4 / 7 = 3 / 7$ , proving the displayed conditional-kernel value. This is the value of the all- $y$ kernel version specified in [F6] at $y = (0, 1)$ ; the theorem does not make conditional-law values pointwise unique.

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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)