This came up in an email exchange regarding a plan to come up with and evaluate Bayesian prediction algorithms for a medical application:
I would not refer to the existing prediction algorithm as frequentist. Frequentist refers to the evaluation of statistical procedures but it doesn’t really say where the estimate or prediction comes from. Rather, I’d say that the Bayesian prediction approach succeeds by adding model structure and prior information.
The advantages of Bayesian inference include:
1. Including good information should improve prediction,
2. Including structure can allow the method to incorporate more data (for example, hierarchical modeling allows partial pooling so that external data can be included in a model even if these external data share only some characteristics with the current data being modeled).
The risks of Bayesian inference include:
3. If the prior information is wrong, it can send inferences in the wrong direction.
4. Bayes inference combines different sources of information; thus it is no longer an encapsulation of a particular dataset (which is sometimes desired, for reasons that go beyond immediate predictive accuracy and instead touch on issues of statistical communication).
OK, that’s all background. The point is that we can compare Bayesian inference with existing methods. The point is not that the philosophies of inference are different—it’s not Bayes vs frequentist, despite what you sometimes hear. Rather, the issue is that we’re adding structure and prior information and partial pooling, and we have every reason to think this will improve predictive performance, but we want to check.
To evaluate, I think we can pretty much do what you say: ROC as basic summary and do graphical exploration, cross-validation (and related methods such as WAIC), and external validation.