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Generalized EM does not decrease the observed log likelihood under Q-ascent

Statement

Assume the Axiom of Choice and the model, fiber, support, integrability, and finite-KL hypotheses of EM does not decrease the observed log likelihood. If an admissible candidate $θ_{t + 1}$ satisfies $Q (θ_{t + 1} ∣ θ_{t}) \geq Q (θ_{t} ∣ θ_{t}),$ then $ℓ_{x} (θ_{t + 1}) \geq ℓ_{x} (θ_{t})$ . The Q maximizer need not be attained for this generalized-EM conclusion.

Facts & Assumptions

Given: AC and the hypotheses of EM does not decrease the observed log likelihood, together with the displayed Q-ascent condition.

[A1]

The Axiom of Choice: the inherited KL identity assumes AC.

[F1]

Expectation-maximization: a generalized update is any admissible candidate whose Q value is at least the current Q value.

[F2]

EM does not decrease the observed log likelihood: on the stated finite, support-compatible fibers, the observed log-likelihood difference equals the Q difference plus a nonnegative directional KL divergence.

Proof

1.1givenA1F2

Apply [F2] to the current and proposed parameters. Its hypotheses hold by the given assumptions, so $ℓ_{x} (θ_{t + 1}) - ℓ_{x} (θ_{t}) = Q (θ_{t + 1} ∣ θ_{t}) - Q (θ_{t} ∣ θ_{t}) + D (q_{t} μ ∥ q_{θ_{t + 1}}^{x} μ)$ .

2.1givenF1F2step 1.1algebra∎

The first term is nonnegative by the Q-ascent condition, and the second is nonnegative by the theorem's KL identity. Hence the observed log likelihood does not decrease. No attained maximum was used.

Depends on

Used by

Likelihood ascent alone gives no global optimality or convergence conclusion Remark

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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)