Unified Approach to Probability Problems and Estimation Algorithms Associated With Symmetric Functions
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Bibliographic record
Abstract
In this article we introduce a simple powerful methodology where we replace the independent variables λ1,...,λnin various symmetric functions as well as in Vieta’s formulas by the indication functions of the events Ai,i = 1,...,n, i.e., λi = 1(Ai),i = 1,...,n. Both the random variable K that counts the number of events that actually occurred and the proposed obvious identity Π_n ^i=1(z−1(Ai)) ≡ (z−1)^KZ^(n−K) that solely depends on K play a central role in this article. Just by choosing different values for z (real, complex, and random) and taking expectations of the various functions we provide other simple proofs of known results as well as obtain new results. The estimation algorithms for computing the expected elementary symmetric functions via least squares based on IFFT in the complex domain (z ∈ C) and least squares or linear programming in the real domain (z ∈ R) are noteworthy. Similarly, we we use Newton’s identities and some well known inequalities to obtain new results and inequalities. Then, we give an algorithm that exactly computes the distribution of K (i.e., q_k:= P(K = k), k = 0,1,...,n) for finite sample spaces. Finally, we give the conclusion and area for further research.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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