On optimal pairwise linear classifiers for normal distributions: the two-dimensional case
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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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 | 0.009 | 0.043 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".