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 finite mixture latent factorization

Example

Let the parameter space be a singleton, let $X$ have sigma-finite reference measure $ν$ , and let the finite latent label set be $Z = {1, \dots, J}$ with counting measure. Choose weights $w_{j} > 0$ with $\sum_{j = 1}^{J} w_{j} = 1$ and probability densities $f_{j}$ with respect to $ν$ . Set $p (x, j) : = w_{j} f_{j} (x) .$ Then the observed marginal density is $m (x) = \sum_{j = 1}^{J} w_{j} f_{j} (x)$ , and on each positive finite fibre the conditional label probabilities are $q (j ∣ x) = w_{j} f_{j} (x) / m (x)$ .

Facts & Assumptions

Given: A finite integer $J \geq 1$ , positive weights summing to one, nonnegative measurable component densities with $\int f_{j} d ν = 1$ , and the singleton parameter space.

[F1]

Latent-variable joint and observed-data factorization: the complete-data density is jointly measurable and integrates to one on each parameter fibre; an observed marginal is obtained by integrating over the latent reference measure.

[F2]

The observed density is the latent marginal under Tonelli: the observed marginal is a density, and division by a positive finite marginal normalizes the latent fibre.

Verification

1.1givenF1algebra

The map $p (x, j) = w_{j} f_{j} (x)$ is nonnegative and product measurable. With counting measure on the finite label set, $\int_{X \times Z} p d (ν \otimes #) = \sum_{j = 1}^{J} w_{j} \int_{X} f_{j} d ν = \sum_{j = 1}^{J} w_{j} = 1.$ It is therefore a complete-data probability density of the type in [F1].

2.1F1F2step 1.1givenalgebra

Integrating over the label coordinate gives $m (x) = \int_{Z} p (x, j) # (d j) = \sum_{j = 1}^{J} w_{j} f_{j} (x)$ . For $0 < m (x) < \infty$ , the conditional label mass is $q (j ∣ x) = w_{j} f_{j} (x) / m (x)$ and its finite sum is $\sum_{j = 1}^{J} q (j ∣ x) = \frac{\sum_{j = 1}^{J} w_{j} f_{j} (x)}{m (x)} = 1.$

3.1step 1.1step 2.1∎

The joint density, its observed marginal, and its normalized positive finite fibres therefore have exactly the finite-mixture factorization claimed.

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)