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Given a composite null hypothesis H, test supermartingales are non-negativesupermartingales with respect to H with initial value 1. Large values of testsupermartingales provide evidence against H. As a result, test supermartingalesare an effective tool for rejecting H, particularly when the p-values obtainedare very small and serve as certificates against the null hypothesis. Examplesinclude the rejection of local realism as an explanation of Bell testexperiments in the foundations of physics and the certification of entanglementin quantum information science. Test supermartingales have the advantage ofbeing adaptable during an experiment and allowing for arbitrary stopping rules.By inversion of acceptance regions, they can also be used to determineconfidence sets. We use an example to compare the performance of testsupermartingales for computing p-values and confidence intervals toChernoff-Hoeffding bounds and the "exact" p-value. The example is the problemof 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.

Performance of Test Supermartingale Confidence Intervals for the Success Probability of Bernoulli Trials