We cordially invite you to the next Bayes club meeting taking place on the 1st of March (Friday) between 4 pm and 5 pm at Leiden University.
Speaker: Peter Grunwald (CWI and Leiden University)
Title: Bayes Factors, S-Values and Optional Continuation
Time: 16:00-17:00, 1st of March, 2019
Location: room 405, MI, Leiden University, Niels Bohrweg 1, Leiden
Abstract: One of the many problems surrounding p-value based null hypothesis testing is this: if our test result is promising but nonconclusive (say, p = 0.07) we cannot simply decide to gather a few more data points. While this “optional continuation” is ubiquitous in science, it invalidates frequentist error guarantees. Bayes factor (BF) hypothesis testing behaves better in this regard, *as long as the null hypothesis is simple*: we can then interpret the BF as (what we call) an ‘S-value’. S-values generalize the concept of nonnegative supermartingale. S-values generically handle optional continuation: if we reject when S > b (say 20), we get a frequentist Type-I error guarantee of 1/b (say 0.05) that *remains valid under optional continuation*. However, if the null is *composite* then the Bayes factor is usually not an S-value and indeed violates Type-I error guarantees. Here we provide a generic solution to this issue – we show that, for arbitrary composite nulls, there exist special priors under which BFs become S-values. In general, these priors are unlike any of the priors encountered in Bayesian practice; however, for the special case where all parameters in the null satisfy a group invariance, using the often-used improper right Haar prior one does get an S-value. Remarkably, Jeffreys (1961) Bayesian t-test, which uses the right Haar on \sigma and a Cauchy on \mu thus gives an S-value and can handle frequentist optional continuation; however, we show that there exists an alternative prior on \mu which gives substantially higher power, while still handling optional continuation.
For the list of upcoming talks and further information about the seminar please visit the seminar website: http://pub.math. leidenuniv.nl/~szabobt/bayes_ club.html