A topologically enriched probability monad on the Cartesian closed category of CGWH spaces
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Bibliographic record
Abstract
Probability monads on categories of topological spaces are classical objects of study in the categorical approach to probability theory, with important applications in the semantics of probabilistic programming languages.We construct a probability monad on the category of compactly generated weakly Hausdorff (CGWH) spaces, a (if not the) standard choice of convenient category of topological spaces.Because a general version of the Riesz representation theorem adapted to this setting plays a fundamental role in our construction, we name it the Riesz probability monad.We show that the Riesz probability monad is a simultaneous extension of the classical Radon and Giry monads that is topologically enriched.Topological enrichment corresponds to a strengthened continuous mapping theorem (in the sense of probability theory).In addition, restricting the Riesz probability monad to the Cartesian closed subcategory of weakly Hausdorff quotients of countably based (QCB) spaces results in a probability monad which is strongly affine, ensuring that the notions of independence and determinism interact as we would expect.We gratefully acknowledge Tobias Fritz, for his encouragement, and his helpful comments in the early stages of this work.Our gratitude also extends to Eveliina Peltola for her thought-provoking impulses.
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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 | 0.005 | 0.001 |
| 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 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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