Modeling Risk with Unit-variance Leptokurtic Fratcal Normal Statistics
Bibliographic record
Abstract
This abstract was created post-production by the JFI Editorial Board. This study examines the hypothesis that there exists a family of unit-variant leptokurtic probability density functions with the attractive properties of normal statistics. It constructs numerically a family of symmetric Normal pdfs defined for fractal spaces with index q. These distributions have the attractive property of having unit variance for 1 <q<2 and approximately unit variance for 0.5<q<1 (the limit of the range that was studied). This paper studies the conjecture that there exists a family of Normal distributions with unit variance that can be defined on metric spaces of dimension m=q/2 for 0<q<2. Some analytical techniques that arise from this conjecture are explored, indicating how one could apply a broader theory of fractal statistics in practice. The power of these techniques follows from the property that the scale of the numerically generated probability distributions is independent of the fractal index q. This allows the non-linearity of fractal statistics to be absorbed into a universal non-linear function of a shape measure with the kurtosis chosen to illustrate the method: for what this study calls Fractal Normal, or Formal, pdfs. It is found that the application of Fractal Normal statistics retains the attractiveness of Normal statistics by allowing optimal solutions to be determined in a straightforward fashion.
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How this classification was reachedexpand
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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".