Classic Probability Revisited (I): Mathematical Models of an Extended Probability Theory
Bibliographic record
Abstract
Part I of this paper presents a set of extended mathematical models of probability theory in order to explain the nature, properties, and rules of general probability. It is found that probability is a hyperstructure beyond those of the traditional monotonic and one-dimensional discrete structures. The sample space of probability is not invariant in general cases. Types of vents in the sample space may be refined as joint or disjoint and dependent, independent, or mutuallyexclusive. These newly identified properties lead to a three-dimensional dynamic model of probability structures constrained by the type of sample spaces, the relation of events, and the dependency of events. A set of algebraic operators on the mathematical structures of the general probability theory is derived based on the extended mathematical models of probability. It is revealed that the Bayes’ law needs to be extended in order to fit more general contexts on variant sample spaces and complex event properties 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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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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 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".