ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

Responsibilities in a two-component mixture

Example

Let $X = R$ with its Borel sigma-algebra and Lebesgue measure, and let latent labels $Z = {1, 2}$ have the power-set sigma-algebra and counting measure. Set $f_{1} (x) = 21_{[0, 1 / 2]} (x)$ and $f_{2} (x) = 1_{[0, 1]} (x)$ ; both are probability densities. For mixture weights $π = (1 / 3, 2 / 3)$ , define the complete-data density $p_{π} (x, j) = π_{j} f_{j} (x)$ . At the observation $x = 1 / 4$ , the observed density is $4 / 3$ and the conditional latent-label probabilities are $(1 / 2, 1 / 2)$ . For any positive candidate weights $λ = (λ_{1}, λ_{2})$ , the EM auxiliary function is the responsibility-weighted complete-data log density.

Facts & Assumptions

Given: The two component densities, current weights $π = (1 / 3, 2 / 3)$ , observation $x = 1 / 4$ , and a candidate weight vector $λ$ with positive coordinates summing to one.

[F1]

Latent-variable joint and observed-data factorization: a jointly measurable complete-data density is normalized over observation and latent coordinates; its observed marginal is the latent-coordinate integral.

[F2]

The observed density is the latent marginal under Tonelli: the observed density is that integral, and its positive finite fibers normalize the quotient to a latent probability density.

[F3]

Bayes formula for dominated kernels: for a proper latent prior and a jointly measurable dominated sampling density, the posterior on a positive finite evidence fiber is prior times likelihood divided by the marginal evidence.

[F4]

Conditional density formula: the normalized quotient on a positive finite fiber is a conditional distribution of the latent coordinate given the observation.

[F5]

Expectation-maximization: $Q (λ ∣ π)$ is the current conditional expectation of $\log p_{λ} (x, Z)$ ; this E-step formula is choice-free, and AC is used only by the KL-based candidate test.

Verification

1.1givenF1F2algebra

Each $f_{j}$ integrates to one, so $p_{π}$ is a probability density on $X \times Z$ because $\sum_{j} π_{j} = 1$ . At $x = 1 / 4$ , $f_{1} (x) = 2$ and $f_{2} (x) = 1$ , hence the observed density is $m_{π} (x) = \sum_{j} π_{j} f_{j} (x) = 4 / 3$ .

2.1givenF2F3step 1.1algebra

The latent conditional weights are $q_{π} (j ∣ x) = π_{j} f_{j} (x) / m_{π} (x)$ by [F2] and [F3], giving $q_{π} (1 ∣ x) = (2 / 3) / (4 / 3) = 1 / 2$ and $q_{π} (2 ∣ x) = (2 / 3) / (4 / 3) = 1 / 2$ .

3.1givenF4F5step 2.1algebra∎

Using this conditional law in the definition [F5] yields $Q (λ ∣ π) = \frac{1}{2} \log (2 λ_{1}) + \frac{1}{2} \log (λ_{2})$ . The two coefficients are exactly the current responsibilities, so this is the complete-data log density averaged over the latent label. By [F4], these weights are also the conditional probabilities for the observation fiber.

Depends on

Used by

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Sources

A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm (standard reference, not scraped)
MIT OpenCourseWare, 18.655 Mathematical Statistics, Lecture 10: Methods of Estimation II (Spring 2016) (standard reference, not scraped)