ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Repeated measurements with correlated noise

Example

Assume AC. Let $θ \sim N (0, 1)$ and observe $Y = A θ + η$ , where $A = (1, 1)^{T}$ and $η \sim N_{2} (0, Γ_{ρ})$ is independent of $θ$ , with $Γ_{ρ} = (\begin{array}{cc} 1 & ρ \\ ρ & 1 \end{array})$ and $- 1 < ρ < 1$ . The matrix $Γ_{ρ}$ is positive definite. The expected information gain is $\frac{1}{2} \log (1 + \frac{2}{1 + ρ})$ . In particular, correlation $ρ = \frac{1}{2}$ gives $\frac{1}{2} \log (7 / 3)$ , less than the independent-noise value $\frac{1}{2} \log 3$ at $ρ = 0$ .

Example references

Chaloner and Verdinelli, §2.2

MacKay, §3.1

Facts & Assumptions

Given: AC, $θ \sim N (0, 1)$ , $Y = A θ + η$ with $A = (1, 1)^{T}$ and independent $η \sim N_{2} (0, Γ_{ρ})$ , and $- 1 < ρ < 1$ .

[A1]

AC is inherited from the EIG/KL convention. The Axiom of Choice

[F1]

For independent $θ \sim N_{n} (m_{0}, C)$ and $η \sim N_{m} (0, Γ)$ with $C ⪰ 0$ and $Γ ≻ 0$ , observing $Y = A θ + η$ has the log-determinant EIG in the Statement of the cited theorem. Linear Gaussian information gain has a log-determinant formula

[F2]

The natural logarithm is strictly increasing. Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

Verification

Given: The hypotheses in the Example.

1.1givenalgebra

The vectors $(1, 1)^{T}$ and $(1, - 1)^{T}$ are eigenvectors of $Γ_{ρ}$ with eigenvalues $1 + ρ$ and $1 - ρ$ . Both are positive for $- 1 < ρ < 1$ , so $Γ_{ρ} ≻ 0$ . With $C = 1$ , independence gives predictive covariance $S_{ρ} = A C A^{T} + Γ_{ρ} = (\begin{array}{cc} 2 & 1 + ρ \\ 1 + ρ & 2 \end{array})$ . Thus $\det S_{ρ} = 4 - (1 + ρ)^{2}$ and $\det Γ_{ρ} = 1 - ρ^{2}$ .

2.1A1F1step 1.1algebra

By [F1], $EIG (ρ) = \frac{1}{2} \log (\det S_{ρ} / \det Γ_{ρ}) = \frac{1}{2} \log (\frac{4 - (1 + ρ)^{2}}{1 - ρ^{2}})$ . Factoring numerator and denominator gives $\frac{4 - (1 + ρ)^{2}}{1 - ρ^{2}} = \frac{(3 + ρ) (1 - ρ)}{(1 - ρ) (1 + ρ)} = 1 + \frac{2}{1 + ρ}$ , proving the formula.

3.1F2step 2.1algebra∎

At $ρ = \frac{1}{2}$ , the ratio is $7 / 3$ ; at $ρ = 0$ , it is $3$ . Since $7 / 3 < 3$ , [F2] gives the strict EIG comparison in the Example.

Depends on

Used by

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Sources

Kathryn Chaloner and Isabella Verdinelli, Bayesian Experimental Design: A Review, Statistical Science 10(3) (1995), 273–304 (standard reference, not scraped)
David J. C. MacKay, Information-Based Objective Functions for Active Data Selection, Neural Computation 4(4) (1992), 590–604 (standard reference, not scraped)