ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

A noisy biological simulator search is conditional on its chosen Gaussian-process model

Example

A biological simulator has two candidate designs $X = (a, b)$ . For this search model, suppose the prior mean is zero, the prior covariance on $(a, b)$ is the identity matrix, the observation noise variance is $1$ , and the acquisition coefficient is $b_{1} = 1$ . Run the procedure with initial list $((a, 0))$ , whose site was fixed before its observation, horizon $N = 2$ , and an observation oracle that returns a finite real value at the selected site. This is a specified surrogate model; no assertion is made that the simulator's actual law is this Gaussian process.

Facts & Assumptions

Given: A two-point Gaussian-process model with $K = I_{2}$ , independent $N (0, 1)$ observation errors, the fixed initial observation $y (a) = 0$ , and the finite-candidate rule in Finite-candidate Gaussian-process acquisition search with explicit statuses.

[F1]

For an observed covariance $C ≻ 0$ , the finite GP posterior mean and covariance are the standard conditional-Gaussian formulas; the observed covariance is the only matrix inverted. Finite Gaussian-process posterior and predictive laws

[F2]

The acquisition rule maximizes posterior mean plus a nonnegative multiple of posterior standard deviation over the finite ordered list. Finite-candidate Gaussian-process acquisition search with explicit statuses

Verification

1.1givenF1algebra

Here $K_{a a} = 1$ , $K_{b a} = 0$ , and $C = K_{a a} + 1 = 2 > 0$ . Conditioning on the observation $y (a) = 0$ gives $μ (a) = 0$ , $v (a) = 1 - 1 / 2 = 1 / 2$ , $μ (b) = 0$ , and $v (b) = 1$ .

1.2givenF2algebra

The two acquisition values are $a_{1} (a) = \sqrt{1 / 2} < 1 = a_{1} (b)$ . The exact least-index finite scan therefore selects $b$ , with maximization error $δ_{1} = 0$ .

2.1step 1.1step 1.2given∎

The procedure calls the observation oracle at $b$ ; if it returns a finite real value, that value is appended. The new observed covariance is $C_{2} = 2 I_{2}$ , so the algorithm computes the posterior at history length $N = 2$ and returns complete. These posterior and acquisition values are conditional on the chosen kernel and noise model. They do not state a regret bound, a global guarantee for the biological simulator, or that the actual simulator is distributed as the surrogate GP.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

Peter I. Frazier, A Tutorial on Bayesian Optimization, §§2–5 (standard reference, not scraped)
Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, Chapter 2 (standard reference, not scraped)