Negative Dependence, Scrambled Nets, and Variance Bounds
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Bibliographic record
Abstract
In this paper, we provide a framework to study the dependence structure of sampling schemes such as those produced by randomized quasi-Monte Carlo methods. The main goal of this new framework is to determine conditions under which the negative dependence structure of a sampling scheme enables the construction of estimators with reduced variance compared to Monte Carlo estimators. To do this, we establish a generalization of the well-known Hoeffding’s lemma—expressing the covariance of two random variables as an integral of the difference between their joint distribution function and the product of their marginal distribution functions—that is particularly well suited to study such sampling schemes. We also provide explicit formulas for the joint distribution of pairs of points randomly chosen from a scrambled (0, m, s)-net. In addition, we provide variance bounds establishing the superiority of dependent sampling schemes over Monte Carlo in a few different setups. In particular, we show that a scrambled (0, m, 2)-net yields an estimator with variance no larger than a Monte Carlo estimator for functions monotone in each variable.
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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