Discrepancy

We have seen that if we have a probability distribution for our model’s parameters, then we can sample from that distribution to see how parameter uncertainty translates into predictive uncertainty. And if we get new data, then we can use Bayes’ Theorem to update the parameter distribution and thereby reduce our predictive uncertainty. So far, so good. But a more difficult problem is that of uncertainty about model structure. We know that all models are wrong, and we discussed model diagnostics in Chap. 10 . But we never know exactly how wrong our models are! In the preceding chapter on Bayesian Model Comparison (Chap. 11 ), we addressed the issue of model structural uncertainty by collecting an ensemble of models, rich enough that we could believe it contained the ‘correct’ model, followed by using data to help us quantify the relative plausibilities of the different models. But what if we are not convinced that our ensemble contains a good model, let alone a correct model? And what if we have only one model that we can work with? In such cases, we need to tackle the issue of model deficiency, or discrepancy, head-on.