Bayesian methods don't perform so much better in the situation you just named. They're also more expensive. The really shine in that they provide a really unif

Bayesian methods don't perform so much better in the situation you just named. They're also more expensive. The really shine in that they provide a really uniform vocabulary of producing new, more sophisticated models while "frequentist" methods usually rely on ingenuity to get to better models. So if you're actively exploring a model space and want to attach a bunch of assumptions and degrees-of-freedom to correspond with a theory you're testing... then Bayesianism is the way to go. Theoretically, marketers are doing that exact process. In practice, they don't see that is a statistical process, though. Maybe someday a tool will bridge that gap successfully... but again, you're unlikely to get a huge advantage with Bayesian methods without that increased work investment.