ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.
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A generalized EM step can increase Q without maximizing it

Example

Let $Θ = (0, 1)$ with its Borel sigma-algebra, and let $X = Z = {0, 1}$ with the power-set sigma-algebra and counting measure. Under parameter $θ$ , take $X, Z$ independent Bernoulli $(θ)$ variables, so $p_{θ} (x, z) = θ^{x + z} (1 - θ)^{2 - x - z}$ . Observe $x = 1$ at current value $θ_{t} = 1 / 2$ . The candidate $θ = 7 / 10$ strictly increases $Q$ but is not its exact maximizer; the observed log likelihood also increases.

Facts & Assumptions

Given: AC for the finite-KL admissibility branch, the displayed two-coordinate Bernoulli model, observation $x = 1$ , current parameter $θ_{t} = 1 / 2$ , and candidate $θ = 7 / 10$ .

[A1]

The Axiom of Choice: AC supplies the Radon–Nikodym branch needed to evaluate finite KL when the current law is absolutely continuous with respect to the candidate law.

[F1]

Latent-variable joint and observed-data factorization: each parameter fiber of a complete-data density integrates to one over observed and latent coordinates.

[F2]

The observed density is the latent marginal under Tonelli: integrating out the latent coordinate gives the observed-data density, and positive finite fibers yield a normalized conditional density.

[F3]

Kullback–Leibler divergence: under AC, $D (P ∥ Q)$ is directional and uses the Radon–Nikodym derivative when $P ≪ Q$ .

[F4]

Expectation-maximization: $Q (θ ∣ θ_{t})$ averages the complete-data log density under the current latent law; exact mode selects an attained maximizer over an admissible candidate set containing the current iterate.

Verification

1.1givenA1F1F2F3F4algebra

For every $θ \in (0, 1)$ , summing $p_{θ} (x, z)$ over $(x, z) \in {0, 1}^{2}$ gives $(θ + (1 - θ))^{2} = 1$ . At $x = 1$ , $m_{θ} (1) = θ$ and the conditional law of $Z$ is Bernoulli $(θ)$ , so the current law is Bernoulli $(1 / 2)$ . Every candidate has positive finite marginal, the same two-point latent support, and $D (q_{t} ∥ q_{θ}) = \frac{1}{2} \log (1 / (2 θ)) + \frac{1}{2} \log (1 / (2 (1 - θ)))$ is finite by [A1, F3]. Their complete-data log values are finite, so every $θ \in (0, 1)$ is admissible by [F4].

2.1givenF2step 1.1algebra

Averaging $\log p_{θ} (1, Z)$ under that current conditional law gives $Q (θ ∣ 1 / 2) = \frac{3}{2} \log θ + \frac{1}{2} \log (1 - θ) = \frac{1}{2} \log (θ^{3} (1 - θ))$ . At $θ = 1 / 2$ , this is $\log (1 / 4)$ ; at $θ = 7 / 10$ , it is $\frac{1}{2} \log (1029 / 10000)$ . Their difference is $\frac{1}{2} \log (1029 / 625) > 0$ , so the candidate strictly raises $Q$ .

3.1givenF4step 2.1algebra∎

For $0 < θ < 1$ , $\frac{27}{256} - θ^{3} (1 - θ) = (θ - 3 / 4)^{2} ((θ + 1 / 4)^{2} + 1 / 8) \geq 0$ , with equality only at $θ = 3 / 4$ . Thus the exact maximizer of $Q$ is $3 / 4$ , so $7 / 10$ is a genuinely nonmaximizing generalized step. The observed log-likelihood difference is $\log (7 / 10) - \log (1 / 2) = \log (7 / 5) > 0$ .

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)