Bounds on the Expectation of a Convex Function of a Random Variable: With Applications to Stochastic Programming
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Bibliographic record
Abstract
This paper is concerned with the determination of tight lower and upper bounds on the expectation of a convex function of a random variable. The classic bounds are those of Jensen and Edmundson-Madansky and were recently generalized by Ben-Tal and Hochman. This paper indicates how still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable. The bounds are applicable in the multivariate case if the random variables are independent. In the dependent case bounds based on the Edmundson-Madansky inequality are not available; however, bounds may be developed using the conditional form of Jensen’s inequality. We give some examples to illustrate the geometrical interpretation and the calculations involved in the numerical determination of the new bounds. Special attention is given to the problem of maximizing a nonlinear program that has a stochastic objective function.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| 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