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.

Information gain on a singular prior support

Example

Assume AC. Let $θ \sim N_{2} (0, C)$ with $C = diag (1, 0)$ , so its support is the first coordinate axis. Let $η \sim N (0, 1)$ be independent of $θ$ and compare the one-coordinate experiments $Y = A θ + η$ for $A_{⊥} = (0, 1)$ and $A_{∥} = (1, 0)$ . Their expected information gains are $0$ and $\frac{1}{2} \log 2$ , respectively. The log-determinant formula remains defined although $C$ is singular.

Example references

Clason, Chapter 13

Dashti and Stuart, §3.1

Facts & Assumptions

Given: AC, $θ \sim N_{2} (0, C)$ with $C = diag (1, 0)$ , $Y = A θ + η$ with independent $η \sim N (0, 1)$ , and the two row operators in the Example.

[A1]

AC is inherited from the finite Gaussian and EIG/KL interfaces. The Axiom of Choice

[F1]

The support of $N_{n} (m, C)$ is $m + Ran (C)$ , including singular $C$ . Support and pseudoinverse coordinates of a finite singular Gaussian

[F2]

For independent $θ \sim N_{n} (m_{0}, C)$ and $η \sim N_{m} (0, Γ)$ with $C ⪰ 0$ and $Γ ≻ 0$ , the linear-Gaussian EIG formula applies to $Y = A θ + η$ , including singular $C$ . Linear Gaussian information gain has a log-determinant formula

[F3]

The natural logarithm is strictly increasing and $\log 1 = 0$ . Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

Verification

Given: The hypotheses in the Example.

1.1A1F1givenalgebra

The range of $C$ is $span {(1, 0)^{T}}$ , so [F1] gives the stated support. Also $A_{⊥} C A_{⊥}^{T} = 0$ and $A_{∥} C A_{∥}^{T} = 1$ .

2.1A1F2step 1.1algebra

Applying [F2] with noise covariance $1$ gives $EIG (A_{⊥}) = \frac{1}{2} \log (1 + 0) = 0$ and $EIG (A_{∥}) = \frac{1}{2} \log (1 + 1) = \frac{1}{2} \log 2$ .

3.1F3step 2.1algebra∎

Since $2 > 1$ , [F3] gives $\log 2 > \log 1 = 0$ . The second coordinate therefore contributes no information under the prior, while the first-coordinate measurement has strictly positive EIG. No inverse or determinant of $C$ is taken.

Depends on

Used by

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Dependency tree · two levels

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Sources

Christian Clason, Regularization of Inverse Problems: Lecture Notes, arXiv:2001.00617v2 (2021) (standard reference, not scraped)
M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems, full lecture notes, arXiv:1302.6989 (standard reference, not scraped)