A Decision Theoretic Approach to A/B Testing

A/B testing is ubiquitous within the machine learning and data science operations of internet companies. Generically, the idea is to perform a statistical test of the hypothesis that a new feature is better than the existing platform---for example, it results in higher revenue. If the p value for the test is below some pre-defined threshold---often, 0.05---the new feature is implemented. The difficulty of choosing an appropriate threshold has been noted before, particularly because dependent tests are often done sequentially, leading some to propose control of the false discovery rate (FDR) rather than use of a single, universal threshold. However, it is still necessary to make an arbitrary choice of the level at which to control FDR. Here we suggest a decision-theoretic approach to determining whether to adopt a new feature, which enables automated selection of an appropriate threshold. Our method has the basic ingredients of any decision-theory problem: a loss function, action space, and a notion of optimality, for which we choose Bayes risk. However, the loss function and the action space differ from the typical choices made in the literature, which has focused on the theory of point estimation. We give some basic results for Bayes-optimal thresholding rules for the feature adoption decision, and give some examples using eBay data. The results suggest that the 0.05 p-value threshold may be too conservative in some settings, but that its widespread use may reflect an ad-hoc means of controlling multiplicity in the common case of repeatedly testing variants of an experiment when the threshold is not reached.