Updating under unknown unknowns: An extension of Bayes’ rule

Developing models to describe observable systems is a challenge because it can be difficult to assess and control the discrepancy between the two entities. We consider the situation of an ensemble of candidate models claiming to accurately describe system features of interest, and ask the question how beliefs about the accuracy of these models can be updated in the light of observations. We show that naive Bayesian updating of these beliefs can lead to spurious results, since the application of Bayes’ rule presupposes the existence of at least one accurate model in the ensemble. We present a framework in which this assumption can be dropped. The basic idea is to extend Bayes’ rule to the exhaustive, but unknown space of all models, and then contract it again to the known set of models by making best/worst case assumptions for the remaining space. We show that this approach leads to an ε-contamination model for the posterior belief, where the ε-contamination is updated along with the distribution of belief across available models. In essence, the ε-contamination provides an additional test on the accuracy of the overall model ensemble compared to the data, and will grow rapidly if the ensemble fails such a test. We demonstrate our concept with an example of auto-regressive processes.