ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Normal precision pooling with two observations

Statement

Assume AC. Let the prior be $μ \sim N (0, 4)$ , and, conditionally on $μ$ , observe independent $Y_{1}, Y_{2} \sim N (μ, 1)$ . For data $(y_{1}, y_{2}) = (1, 3)$ , the evidence is finite and positive and the posterior is $N (16 / 9, 4 / 9)$ .

Facts & Assumptions

Given: AC, the stated Gaussian model, and observed data $(1, 3)$ .

[A1]

The Axiom of Choice is required by the Gaussian density normalization used in the update theorem. The Axiom of Choice.

[F1]

For a proper prior $N (μ_{0}, τ_{0}^{2})$ and $n \geq 1$ conditionally IID $N (μ, σ^{2})$ data, the evidence is finite positive and the posterior variance and mean are $V_{n} = (τ_{0}^{- 2} + n σ^{- 2})^{- 1}$ and $V_{n} (μ_{0} / τ_{0}^{2} + n \overset{ˉ}{y} / σ^{2})$ . Known-variance Normal-Normal conjugate update.

Verification

Given: The hypotheses and data in the Statement, with [A1] and [F1].

1.1givenalgebra

The sample mean is $\overset{ˉ}{y} = (1 + 3) / 2 = 2$ . The prior precision is $1 / 4$ and the two observations contribute total precision $2 / 1 = 2$ , so the posterior precision is $9 / 4$ .

2.1A1F1step 1.1algebra∎

Thus $V_{2} = (9 / 4)^{- 1} = 4 / 9$ , and the posterior mean is $V_{2} (0 / 4 + 2 \cdot 2 / 1) = (4 / 9) 4 = 16 / 9$ . Applying [F1] gives posterior $N (16 / 9, 4 / 9)$ and finite positive evidence.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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

Gelman et al., Bayesian Data Analysis, 3rd ed., §2.5 (standard reference, not scraped)
Johnson, Ott & Dogucu, Bayes Rules!, Chapter 5, §5.3 (standard reference, not scraped)