Subjective Probability and Geometry: Three Metric Theorems Concerning Random Quantities
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
Affine properties are more general than metric ones because they are independent of the choice of a coordinate system. Nevertheless, a metric, that is to say, a scalar product which takes each pair of vectors and returns a real number, is meaningful when $n$ vectors, which are all unit vectors and orthogonal to each other, constitute a basis for the $n$-dimensional vector space $\mathcal{A}$. In such a space $n$ events $E_i$, $i = 1, \ldots, n$, whose Cartesian coordinates turn out to be $x^i$, are represented in a linear form. A metric is also meaningful when we transfer on a straight line the $n$-dimensional structure of $\mathcal{A}$ into which the constituents of the partition determined by $E_1, \ldots, E_n$ are visualized. The dot product of two vectors of the $n$-dimensional real space $\mathbb{R}^n$ is invariant: of these two vectors the former represents the possible values for a given random quantity, while the latter represents the corresponding probabilities which are assigned to them in a subjective fashion.We deduce these original results, which are the foundation of our next and extensive study concerning the formulation of a geometric, well-organized and original theory of random quantities, from pioneering works which deal with a specific geometric interpretation of probability concept, unlike the most part of the current ones which are pleased to keep the real and deep meaning of probability notion a secret because they consider a success to give a uniquely determined answer to a problem even when it is indeterminate.Therefore, we believe that it is inevitable that our references limit themselves to these pioneering works.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.027 | 0.077 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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