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.

Posterior prediction adds parameter uncertainty to future sampling variation

Example

Let the posterior put mass $1 / 2$ on each of $θ = 0$ and $θ = 1$ , and let the future kernel be $Q_{θ} = Bernoulli (θ)$ . The posterior-predictive law is Bernoulli with parameter $1 / 2$ .

Facts & Assumptions

Given: $Π ({0}) = Π ({1}) = 1 / 2$ and, conditionally on $θ \in {0, 1}$ , the future observation has law $Bernoulli (θ)$ .

[F1]

The posterior-predictive law averages future kernels over the posterior. Posterior-predictive probability kernel

[F2]

Its finite mean and variance decompose into averaged conditional moments and the posterior variance of the conditional mean. Posterior-predictive mean and variance decomposition

Verification

Given: The hypotheses in the Example and Facts & Assumptions.

1.1givenF1algebra

By [F1], $Q^{p r e d} ({1}) = \frac{1}{2} Q_{0} ({1}) + \frac{1}{2} Q_{1} ({1}) = \frac{1}{2} (0) + \frac{1}{2} (1) = \frac{1}{2}$ . Since its support is ${0, 1}$ , this is $Bernoulli (1 / 2)$ , with mean $1 / 2$ and variance $1 / 4$ .

1.2givenalgebra

The conditional means are $μ (0) = 0$ and $μ (1) = 1$ , and both conditional variances are zero. Thus $E_{Π} [Var (Z ∣ θ)] = 0$ and ${Var}_{Π} (μ) = 1 / 4$ .

2.1F2step 1.1step 1.2∎

The predictive variance in step 1.1 equals the sum of those two terms in step 1.2, as asserted by [F2]. Here the entire predictive variance comes from posterior uncertainty about $θ$ , while future sampling conditional on $θ$ contributes zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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 et al., Bayesian Data Analysis, 3rd ed., §2.5 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 8 §8.3 (standard reference, not scraped)