Classic Probability Revisited (II): Algebraic Operations of the Extended Probability Theory
Why this work is in the frame
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Bibliographic record
Abstract
Part II of this paper presents a set of comprehensive algebraic operators on the extended mathematical structures of the general probability theory. It is recognized that the classic probability theory is cyclically defined among a small set of highly coupled operations. In order to solve this fundamental problem, a reductive framework of the general probability theory is introduced. It is found that conditional probability operation on consecutive events is the key to independently manipulate other probability operations. This leads to a revisited framework of rigorous manipulations on general probabilities. It also provides a proof for a revisited Bayes’ law fitting in more general contexts of variant sample spaces and complex event relations in fundamental probability theories. The revisited probability theory enables a rigorous treatment of uncertainty events and causations in formal inference, qualification, quantification, and semantic analysis in contemporary fields such as cognitive informatics, computational intelligence, cognitive robots, complex systems, soft computing, and brain informatics.
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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 | 0.002 | 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 | 0.001 | 0.002 |
| 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