Formalization of Measure Theory and Lebesgue Integration for Probabilistic Analysis in HOL
Bibliographic record
Abstract
Dynamic systems that exhibit probabilistic behavior represent a large class of man-made systems such as communication networks, air traffic control, and other mission-critical systems. Evaluation of quantitative issues like performance and dependability of these systems is of paramount importance. In this paper, we propose a generalized methodology to formally reason about probabilistic systems within a theorem prover. We present a formalization of measure theory in the HOL theorem prover and use it to formalize basic concepts from the theory of probability. We also use the Lebesgue integration to formalize statistical properties of random variables. To illustrate the practical effectiveness of our methodology, we formally prove classical results from the theories of probability and information and use them in a data compression application in HOL.
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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 | 0.009 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.009 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.001 | 0.005 |
| 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".