ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

Integrating hyperparameter uncertainty versus plugging in

Example

Let the hyperparameter be $t = τ^{2} \in {1, 4}$ with prior probability $1 / 2$ at each value. There is one group, $θ ∣ t \sim N (0, t)$ , and $Y ∣ θ \sim N (θ, 1)$ . Observe $y = 0$ . Specify the likelihood-density version pointwise by $f (θ, y) = (2 π)^{- 1 / 2} \exp (- (y - θ)^{2} / 2) (θ, y \in R) .$ The full posterior for $θ$ is a mixture of the two conditional normal posteriors. Plugging in the posterior mode of $t$ instead yields just one component, which is not the full posterior.

Facts & Assumptions

Given: A proper two-point hyperprior on $t \in {1, 4}$ , one group parameter $θ ∣ t \sim N (0, t)$ , an observation kernel $Y ∣ θ \sim N (θ, 1)$ , and observed value $y = 0$ .

[F1]

The hierarchy's prior and observation kernels define the joint latent-data model. Finite-group hierarchical Bayesian model

[F2]

On positive finite evidence, the joint posterior is the normalized prior-times-likelihood measure. Hierarchical posterior normalization on positive finite evidence

[F3]

With the canonical normal likelihood density specified in the Example and positive conditional variances, the group posterior conditional on fixed normal hyperparameters is the normal law from the Normal–Normal update. Conditional normal shrinkage in a finite hierarchy

[F4]

For $a \in R$ and $v > 0$ , the $N (a, v)$ density is $(2 π v)^{- 1 / 2} e^{- (x - a)^{2} / (2 v)}$ . A positive-variance normal law has its location-scale density

[F5]

The standard normal density has Lebesgue integral one. The standard normal density has total mass one

[A1]

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

Verification

Given: The model and observed value in the Example.

1.1A1F1F4F5givenalgebra

The specified model supplies the prior and sampling kernels [F1]; their densities are given by [F4]. For fixed $t > 0$ , multiply the two densities and integrate over $θ$ . The integrand is $(2 π \sqrt{t})^{- 1} \exp (- (1 + t^{- 1}) θ^{2} / 2)$ . Substituting $z = \sqrt{1 + t^{- 1}} θ$ and using [F5] for the unit normal integral gives the marginal evidence $m_{t} (0) = 1 / \sqrt{2 π (1 + t)}$ , which is positive and finite for $t = 1, 4$ .

2.1F2step 1.1algebra

The two prior masses are equal, so [F2] gives posterior odds $P (t = 1 ∣ y = 0) / P (t = 4 ∣ y = 0) = m_{1} (0) / m_{4} (0) = \sqrt{5 / 2} > 1$ . Thus the posterior mode is $t = 1$ , and the posterior weights $w = P (t = 1 ∣ y = 0)$ and $1 - w$ are both strictly between zero and one.

3.1A1F3step 2.1algebra

By [F3], $θ ∣ t = 1, y = 0 \sim N (0, 1 / 2)$ and $θ ∣ t = 4, y = 0 \sim N (0, 4 / 5)$ . The full posterior is their mixture with weights $w$ and $1 - w$ , so its second moment is $w / 2 + (1 - w) 4 / 5 > 1 / 2$ .

4.1step 3.1algebra∎

Plugging in the posterior mode $t = 1$ gives only $N (0, 1 / 2)$ , whose second moment is $1 / 2$ . This differs from the full posterior's second moment in step 3.1, so the plug-in posterior is not the full posterior mixture.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 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

Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, 3rd ed., §§5.3–5.4 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 16 (standard reference, not scraped)