Is a spike-and-slab prior ever a good idea?

Statisticians who use Bayes factors often complain that, somewhat awkwardly, their hypothesis testing and estimation results can disagree: a Bayes factor may favour the null hypothesis while the 95% credible interval excludes the null value. In this talk, I show that this apparent contradiction disappears once testing and estimation are required to use the same "implied prior model odds." Upon careful consideration, we see that using a Bayes factor for decision-making implicitly commits one to a mixture model and, when one's null is a point null, implicitly specifies a spike-and-slab prior. Such priors are not without their challenges. The resulting model-averaged posterior has an atom at the null value, its CDF has a jump, and credible intervals at conventional levels such as 95% may not exist at all. Moreover, if the frequentist p-value is held fixed while the sample size grows, the range of probability levels at which a credible interval can be defined steadily shrinks. This is a lesser-known correlate of the Jeffreys–Lindley paradox, one that carries the paradox from testing into interval estimation. All of this leads to a simple question: is a spike-and-slab prior (or, equivalently, a Bayes factor against a point null) ever a good idea?  

To join this seminar virtually, please request Zoom connection details from hr.ops@stat.ubc.ca.