Univariate marginal distribution algorithm

⚠️ Curated Limitations & Negative Results

• This record covers the original unsmoothed binary procedure. Probability margins, mutation, smoothing, restarts, nonbinary distribution families and later runtime-analysis variants change or supplement that procedure. The skeleton does not completely specify selection multiplicities, tie handling, initialization probabilities or termination.
• A product sampling law does not guarantee independence of the empirical coordinates in every finite sample, or preserve dependencies in selected candidates. More expressive conditional models are separate constructions; the original article expressly leaves CDA incomplete.
• The proportionate-selection equations require nonnegative fitness and positive mean fitness. They cannot be applied unchanged to arbitrary signed objectives, zero mean fitness, truncation selection or a particular finite sampled generation. The chapter’s continuous expected-fitness interpretation does not certify global optimization: on a two-bit landscape with f(00)=f(11)=2 and f(01)=f(10)=1, p_1=p_2=1/2 is stationary under the ideal recurrence despite a larger attainable mean fitness at a maximizer. That stationary law still samples optimal strings with positive probability.
• The 1996 SUMDA fixed-selected-frequency result assumes r(t)=c; it is not a general adaptive-convergence theorem when selected frequencies change. The basic blend is cited from its update equation. The later chapter’s function-specific finite-population examples and conjectural sample-size discussion are not universal bounds.
• Formula-level overlap does not establish full-method equivalence. Under matching selection and tie policies, an unsmoothed Bernoulli cross-entropy refit can produce the same next distribution. Full incremental PBIL with mutation, natural evolution strategies and information-geometric optimization have distinct maintained scopes. No benchmark, author implementation or optimizer run was used to establish superiority or a convergence rate.
• The primary 1996 paper was read from an author-hosted ten-page copy using OCR and selected formula visuals; its file build date is separate from publication. The later 39-page author manuscript is indexed as 2000 but associated with the 2001 published chapter. Its reading PDF is an explicitly documented derivative of the retained original PostScript after removal of a trailing MIME delimiter following the document EOF. The source hashes and conversion failures preserve that distinction; publisher-byte equivalence is not asserted.