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Suppose that we are given sample access to an unknown distribution p over n elements and an explicit distribution q over the same n elements. We would like to reject the null hypothesis“p=q” after seeing as few samples as possible, whenp6=q, while we never want to reject the null, when p=q. Well-known results show thatΘ(√n/2)samples are necessary and sufficient for distinguishing whether p equals q versus p is-far from q in total variation distance. However,this requires the distinguishing radiusto be fixed prior to deciding how many samples to request.Our goal is instead to design sequential hypothesis testers, i.e. online algorithms that request i.i.d.samples from p and stop as soon as they can confidently reject the hypothesis p=q, without being given a lower bound on the distance between p and q, whenp6=q. In particular, we want to minimize the number of samples requested by our tests as a function of the distance between p and q, and if p=q we want the algorithm, with high probability, to never reject the null. Our work is motivated by and addresses the practical challenge of sequential A/B testing in Statistics.We show that, when n= 2, any sequential hypothesis test must seeΩ(1dtv(p,q)2log log1dtv(p,q))samples, with high (constant) probability, before it rejects p=q, where dtv(p,q) is the—unknown to the tester—total variation distance between p and q. We match the dependence of this lower bound ondtv(p,q)by proposing a sequential tester that rejects p=q from at most O(√ndtv(p,q)2log log1dtv(p,q))samples with high (constant) probability. TheΩ(√n)dependence on the support size n is also known to be necessary. We similarly provide two-sample sequential hypothesis testers, when sample access is given to both p and q, and discuss applications to sequential A/B testing.
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Sequential Monte Carlo testing by betting
Abstract In a Monte Carlo test, the observed dataset is fixed, and several resampled or permuted versions of the dataset are generated in order to test a null hypothesis that the original dataset is exchangeable with the resampled/permuted ones. Sequential Monte Carlo tests aim to save computational resources by generating these additional datasets sequentially one by one and potentially stopping early. While earlier tests yield valid inference at a particular prespecified stopping rule, our work develops a new anytime-valid Monte Carlo test that can be continuously monitored, yielding a p-value or e-value at any stopping time possibly not specified in advance. It generalizes the well-known method by Besag and Clifford, allowing it to stop at any time, but also encompasses new sequential Monte Carlo tests that tend to stop sooner under the null and alternative without compromising power. The core technical advance is the development of new test martingales for testing exchangeability against a very particular alternative based on a testing by betting technique. The proposed betting strategies are guided by the derivation of a simple log-optimal betting strategy, have closed-form expressions for the wealth process, provable guarantees on resampling risk, and display excellent power in practice.
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- PAR ID:
- 10581195
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 87
- Issue:
- 4
- ISSN:
- 1369-7412
- Format(s):
- Medium: X Size: p. 1200-1220
- Size(s):
- p. 1200-1220
- Sponsoring Org:
- National Science Foundation
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