{
  "schema_version": 1,
  "problem_number": "AIM-COMPUTATION-0025",
  "title": "Exponential Rarity of Multiple Likelihood Modes in Bivariate Seemingly Unrelated Regression",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The likelihood of the crossed bivariate Gaussian seemingly unrelated regression model can have several local modes. Its almost-sure eventual uniqueness is known. We give an explicit finite-sample bound: for fixed noncollinear regressor vectors of correlation r, with d=1-|r| and k=n-2>=2, the probability of more than one stationary point is at most exp[-k d^3/(16(1+|r|))]+2 exp[-k/16]. The bound is uniform over slopes and all positive-definite error covariances. A determinant normal form gives a directly checkable unimodality certificate, which combines with elementary Gaussian tails. Common regressors are allowed by replacing k with n-p-2 after projection. For Gaussian random predictors of population correlation r0, an explicit bound is 11 exp[-(n-2)(1-|r0|)^3/512]. A change-of-measure argument shows that the probability of multiple modes is exp[-Theta(n)] at each fixed nonsingular random-design parameter. Constants are not claimed optimal. A classical algebraic MANOVA certificate distinguishes the two parts of the motivating AIM question.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "keywords": [
    "seemingly unrelated regression",
    "likelihood multimodality",
    "finite-sample bound",
    "Gaussian regression",
    "exponential concentration",
    "algebraic statistics",
    "AIM-COMPUTATION-0025",
    "math.ST",
    "stat.TH"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-computation-0025/",
  "pdf_url": "https://eulersolve.org/papers/aim-computation-0025/paper.pdf?v=1dd78d1ef7a7",
  "doi": "10.5281/zenodo.23032095",
  "zenodo_record_url": "https://zenodo.org/records/23032095",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Correctly specified Gaussian crossed bivariate model only; no optimal exponent, exact probability or absolute priority. MANOVA is classical. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
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      "sha256": "1dd78d1ef7a7073193f80738ccdd44ae9041652dc42cfc7fa88091f42ce49212"
    },
    "source.zip": {
      "sha256": "9a9fc70659bb649ee20e7af6da5404580d35ba7eef84ea32c7d627a47c1b0c2a"
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    "verification_report.md": {
      "sha256": "2d2a17c69654b831f55e03a0d6d262c7f3452fbd3ec87fb663feb18301f9e206"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}