The multivariate Selberg beta distribution and applications
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Bibliographic record
Abstract
Abstract The classical beta distribution defined on (0,1) is based on Euler's integral of the second type and its generalization to several variables, defined on a simplex, is based on Dirichlet's generalized form of Euler's integral. We define the multivariate Selberg beta random vector X=(X 1, …, X p ) ∼ MSBeta1 (α, β, γ, p), p≥1, defined on (0, 1) p . This distribution, based on the Selberg integral, a generalized form of the Dirichlet integral, has close relationships with the distributions of roots of determinantal equations and also has several interesting applications, such as the multivariate Gini mean difference. Keywords: Selbergbetarandom vectordistancemultivariate Gini mean differencesample dispersion measure AMS Subject Classification : 62E9962H10 Acknowledgements Research supported in part by NSERC grant A8249(Canada). The author wishes to thank Francis Niyomwungere for his excellent work in programming complex expressions.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it