Certifying Maximum Likelihood Degrees of Matroid Strata
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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.<br> <br> 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 <strong>lower bound</strong> for the ML degree of the corresponding matroid. <br> 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]. <strong>References</strong> [1] D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels, and S. Telen: <em>Likelihood Degenerations</em>, arXiv:2107.10518. [2] P. Breiding, K. Rose, and S. Timme: <em>Certifying zeros of polynomial systems using interval arithmetic</em>, arXiv:2011.05000.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| 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