Classic Probability Revisited (I): Mathematical Models of an Extended Probability Theory
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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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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 | 0.003 | 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 | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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