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.

A support-losing candidate has infinite directional KL

Example

Let $Θ = [0, 1]$ with its Borel sigma-algebra, let the observation space be the singleton ${x_{0}}$ with counting measure, and let the latent space be ${0, 1}$ with counting measure. Set $p_{θ} (x_{0}, 1) = θ$ and $p_{θ} (x_{0}, 0) = 1 - θ$ . At current value $θ_{t} = 1 / 2$ , the latent law is $(1 / 2, 1 / 2)$ ; candidate $θ = 0$ has latent law $(1, 0)$ . The observed marginal is one at both parameters, but the current-to-candidate KL divergence is infinite. The extended diagnostic integral of $\log p_{0} (x_{0}, Z)$ under the current latent law is $- \infty$ when $\log 0 = - \infty$ ; the algorithm's $Q (0 ∣ 1 / 2)$ is undefined because its Definition assigns $Q$ only when the integral is finite.

Facts & Assumptions

Given: The displayed finite complete-data model, fixed observation $x_{0}$ , current value $θ_{t} = 1 / 2$ , candidate value $θ = 0$ , and the convention $\log 0 = - \infty$ .

[F1]

Latent-variable joint and observed-data factorization: a complete-data density is normalized over the observation and latent coordinates for every parameter.

[F2]

Kullback–Leibler divergence: if the first probability is not absolutely continuous with respect to the second, the directional divergence is $+ \infty$ ; this branch uses no Radon–Nikodym representative or Axiom of Choice.

[F3]

Expectation-maximization: Q is defined only when its integral is finite; admissibility requires finite directional KL and integrability of the candidate log density, and the $q_{t} ≪̸ q_{θ}$ rejection branch is choice-free while the $q_{t} ≪ q_{θ}$ finite-KL branch assumes AC.

Verification

1.1givenF1algebra

For each $θ \in [0, 1]$ , the complete-data masses sum to $θ + (1 - θ) = 1$ , and the observed marginal is $m_{θ} (x_{0}) = 1$ . The current and candidate latent laws are therefore $q_{t} = (1 / 2, 1 / 2)$ and $q_{0} = (1, 0)$ .

2.1givenF2F3step 1.1algebra

The singleton set ${1}$ has $q_{0}$ -mass zero and $q_{t}$ -mass $1 / 2$ , so $q_{t} ≪̸ q_{0}$ and $D (q_{t} ∥ q_{0}) = + \infty$ by [F2]. The extended diagnostic integral is $\frac{1}{2} \log (1 - 0) + \frac{1}{2} \log 0 = - \infty$ ; it is not a finite value of $Q (0 ∣ 1 / 2)$ under [F3].

3.1F3step 2.1∎

The observed log likelihood is zero at both parameters, but the candidate violates the support-compatible admissibility condition of [F3]. It is not an update allowed by the defined procedure.

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