Performance of Test Supermartingale Confidence Intervals for the Success Probability of Bernoulli Trials
Bibliographic record
Abstract
Given a composite null hypothesis H0, test supermartingales are non-negative supermartingales with respect to H0 with an initial value of 1. Large values of test supermartingales provide evidence against H0. As a result, test supermartingales are an effective tool for rejecting H0, particularly when the p-values obtained are very small and serve as certificates against the null hypothesis. Examples include the rejection of local realism as an explanation of Bell test experiments in the foundations of physics and the certification of entanglement in quantum information science. Test supermartingales have the advantage of being adaptable during an experiment and allowing for arbitrary stopping rules. By inversion of acceptance regions, they can also be used to determine confidence sets. We used an example to compare the performance of test supermartingales for computing p-values and confidence intervals to Chernoff-Hoeffding bounds and the “exact” p-value. The example is the problem of inferring the probability of success in a sequence of Bernoulli trials. There is a cost in using a technique that has no restriction on stopping rules, and, for a particular test supermartingale, our study quantifies this cost.
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How this classification was reachedexpand
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 | 0.005 | 0.003 |
| 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 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".