Estimating means of bounded random variables by betting
Ian Waudby-Smith, Aaditya Ramdas
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Abstract We derive confidence intervals (CIs) and confidence sequences (CSs) for the classical problem of estimating a bounded mean. Our approach generalizes and improves on the celebrated Chernoff method, yielding the best closed-form "empirical-Bernstein" CSs and CIs (converging exactly to the oracle Bernstein width) as well as non-closed-form "betting" CSs and CIs. Our method combines new composite nonnegative (su ...
per)martingales with Ville's maximal inequality, with strong connections to testing by betting and the method of mixtures. We also show how these ideas can be extended to sampling without replacement. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform prior approaches, establishing a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, when sampling with and without replacement.
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Abstract We derive confidence intervals (CIs) and confidence sequences (CSs) for the classical problem of estimating a bounded mean.
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Utility signals: depth 60/100, grounding 58/100, status medium.
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Research context
80
Citations
94
References
Tasks
Bounded function, Oracle, Variance (accounting), Sampling (signal processing), Chernoff bound, Random variable, Statistics, Computer science
Methods
Mathematical optimization
Domains
Mathematics, Applied mathematics
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