ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

A zero-normalizer latent fiber

Example

Take $X = Z = {0, 1}$ with counting measure and a singleton parameter. Define the joint density by $p (0, 0) = 1$ and $p (x, z) = 0$ for the other three pairs. The complete-data law is normalized, but the observed fibre at $x = 1$ has marginal zero.

Facts & Assumptions

Given: The finite observation and latent spaces with counting reference measures, a singleton parameter space, and the stated function $p$ .

[F1]

Latent-variable joint and observed-data factorization: a complete-data density must integrate to one; the ratio $p (x, z) / m (x)$ is defined as a conditional density only when $0 < m (x) < \infty$ .

[F2]

The observed density is the latent marginal under Tonelli: the observed marginal is $m (x) = \sum_{z} p (x, z)$ .

Verification

1.1givenF1algebra

The only nonzero summand is $p (0, 0) = 1$ , so $\sum_{x, z} p (x, z) = 1$ ; this is a normalized joint density on the finite product space.

1.2F2givenalgebra

Summing over $z$ gives $m (0) = 1$ and $m (1) = 0$ . At $x = 1$ , both $p (1, z)$ and $m (1)$ are zero, so the quotient $p (1, z) / m (1)$ is undefined.

2.1F1step 1.2∎

Since $m (1) = 0$ , the observation $x = 1$ is outside the positive finite fibre set in [F1]. The formula supplies no pointwise conditional latent density there.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Mario A. T. Figueiredo, Lecture Notes on the EM Algorithm (2004) (standard reference, not scraped)