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.

Complete, partial, and no-pooling limits in the normal hierarchy

Example

In the known-variance normal hierarchy, fix $μ$ , an observed group summary ${\overset{ˉ}{y}}_{j}$ , and $s_{j}^{2} > 0$ . Specify the likelihood-density version pointwise by $f_{j} (θ, y) = (2 π s_{j}^{2})^{- 1 / 2} \exp ⁣ (- \frac{(y - θ)^{2}}{2 s_{j}^{2}}) (θ, y \in R) .$ For finite $τ^{2} > 0$ , the posterior mean is a strictly weighted average of ${\overset{ˉ}{y}}_{j}$ and $μ$ . As $τ^{2} ↓ 0$ it approaches complete pooling at $μ$ ; at the boundary $τ^{2} = 0$ , the prior itself forces $θ_{j} = μ$ . As $τ^{2} \to \infty$ , the posterior mean and variance approach the group-only values ${\overset{ˉ}{y}}_{j}$ and $s_{j}^{2}$ , the no-pooling limit. The value $τ^{2} = \infty$ is not an admissible proper prior.

Facts & Assumptions

Given: A finite group hierarchy, fixed $μ$ , observed ${\overset{ˉ}{y}}_{j}$ , known $s_{j}^{2} > 0$ , and conditional variance $τ^{2} > 0$ except when examining the stated boundary limit.

[F1]

For positive $τ^{2}, s_{j}^{2}$ and the canonical normal likelihood-density version specified in the Example, the conditional posterior mean and variance are ${\hat{θ}}_{j} = ({\overset{ˉ}{y}}_{j} / s_{j}^{2} + μ / τ^{2}) / (1 / s_{j}^{2} + 1 / τ^{2})$ and $V_{j} = (1 / s_{j}^{2} + 1 / τ^{2})^{- 1}$ . Conditional normal shrinkage in a finite hierarchy

[F2]

The library normal law includes the degenerate boundary $N (μ, 0) = δ_{μ}$ . Standard normal and normal laws

[A1]

AC is assumed for the library normal-law convention. The Axiom of Choice

[A2]

$τ^{2} > 0$ and $s_{j}^{2} > 0$ for the conditional-posterior formula.

Verification

Given: The normal hierarchy in the Example and Facts & Assumptions.

1.1A1A2F1algebra

Set $a_{j} = τ^{2} / (τ^{2} + s_{j}^{2})$ . Algebra in [F1] gives ${\hat{θ}}_{j} = a_{j} {\overset{ˉ}{y}}_{j} + (1 - a_{j}) μ$ and $V_{j} = τ^{2} s_{j}^{2} / (τ^{2} + s_{j}^{2})$ . Since both variances are positive, $0 < a_{j} < 1$ , so for finite positive $τ^{2}$ the group mean uses both its own data and the population mean.

2.1A1A2F1F2step 1.1algebra

As $τ^{2} ↓ 0$ , $a_{j} \to 0$ , $V_{j} \to 0$ , and ${\hat{θ}}_{j} \to μ$ . At $τ^{2} = 0$ , [F2] makes the conditional prior $δ_{μ}$ , so every group parameter equals $μ$ and conditioning on the data leaves it there. This is complete pooling.

3.1A1A2F1step 1.1algebra∎

As $τ^{2} \to \infty$ , $a_{j} \to 1$ , $V_{j} \to s_{j}^{2}$ , and ${\hat{θ}}_{j} \to {\overset{ˉ}{y}}_{j}$ . These are the group-only posterior mean and variance, so they describe the no-pooling limit. No posterior at $τ^{2} = \infty$ is asserted.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, 3rd ed., §5.4 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapters 15–16 (standard reference, not scraped)