A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications
Why this work is in the frame
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Bibliographic record
Abstract
Our main, novel, result is that a certain non‐central chi‐squared mixture of non‐central chi‐squared distributions is itself a scaled non‐central chi‐squared distribution. From this and a link to a known result on a mixture representation for a scaled central chi‐squared distribution, numerous further mixture results, both old and new, ensue. These include mixture results for central F , non‐central F and Libby–Novick distributions. The main result involves distributions all with the same degrees of freedom; it is also extended to the case where the mixing non‐central chi‐squared distribution has degrees of freedom an even number larger than that of the conditional non‐central chi‐squared distribution, with further consequences pursued.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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