Certifying Maximum Likelihood Degrees of Matroid Strata
Bibliographic record
Abstract
We present the result of the certification method described in [1] for the computation of the ML degree for matroids of rank k on m elements. The main theorem in Section 4 of [1], for fixed k, m, illustrates the matroid strata by dimension, and all occurring ML degrees together with their multiplicity of occurrences. In each folder 'CertificatesAndSummaries'*k*m, for each simple rank k matroid on m atoms indexed by the database we used (with the exceptions of (3,9) #1#2#3#5 and (4,8) #1#2) there are two files: MatroidSummary_i and Certificate_i. MatroidSummary_i summarizes the certification attempts we made - that is, how many, and what the certified lower bound for the ML degree was. An instance of a certification which obtained the maximum bound is saved in Certificate_i. For some matroids, we had the resources to run the certification process multiple times, whereas for matroids with large ML degrees, we only ran it once. Nonetheless, each certificate produces a lower bound for the ML degree of the corresponding matroid. If an additional index exists, this indicates that the realization space of the matroid had multiple irreducible components (see (4,8) # 160 which has two components) and each file corresponds to the above process for a single irreducible component. Finally, the file Certificate_48 certifies the lower bound of the Euler characteristic of the space X(4,8) discussed in Section 6 of [1]. References [1] D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels, and S. Telen: Likelihood Degenerations, arXiv:2107.10518. [2] P. Breiding, K. Rose, and S. Timme: Certifying zeros of polynomial systems using interval arithmetic, arXiv:2011.05000.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.047 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".