On optimal pairwise linear classifiers for normal distributions: the two-dimensional case
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Bibliographic record
Abstract
Optimal Bayesian linear classifiers have been studied in the literature for many decades. We demonstrate that all the known results consider only the scenario when the quadratic polynomial has coincident roots. Indeed, we present a complete analysis of the case when the optimal classifier between two normally distributed classes is pairwise and linear. We focus on some special cases of the normal distribution with nonequal covariance matrices. We determine the conditions that the mean vectors and covariance matrices have to satisfy in order to obtain the optimal pairwise linear classifier. As opposed to the state of the art, in all the cases discussed here, the linear classifier is given by a pair of straight lines, which is a particular case of the general equation of second degree. We also provide some empirical results, using synthetic data for the Minsky's paradox case, and demonstrated that the linear classifier achieves very good performance. Finally, we have tested our approach on real life data obtained from the UCI machine learning repository. The empirical results that we obtained show the superiority of our scheme over the traditional Fisher's discriminant classifier.
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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 | 0.001 | 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)
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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