Alphabeta Math
RemarkRemark: AI-adaptedProof: Not applicablePipeline-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.

Likelihood ascent alone gives no global optimality or convergence conclusion

Remark

Under the hypotheses of EM does not decrease the observed log likelihood and Generalized EM does not decrease the observed log likelihood under Q-ascent, including AC, these are one-step inequalities for a fixed data fiber. They do not prove that the likelihood sequence is bounded, that parameter iterates converge, that a limit point is stationary, or that a stationary point is globally optimal. The exact M-step also requires an attained Q maximum; ascent itself does not supply one. These are limits of the stated ascent result, not additional convergence theorems.

Remarks

The examples An EM likelihood with two strict local maxima ↗ and Gaussian mixture likelihood is unbounded under covariance collapse ↗ exhibit tied maxima and an unbounded observed likelihood.

Depends on

Used by

Dependency tree · two levels

15 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