Uncertainty quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball

Uncertainty quantification (UQ) is, broadly, the task of determining appropriate uncertainties to model predictions. There are essentially three kinds of approaches to Uncertainty Quantification: (A) robust optimization (min and max), (B) Bayesian (conditional average) and (C) decision theory (minmax). Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally non-robust (brittle) with respect to the choice of that prior and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of loss/risk used to identify the prior is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample x, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties: (a) an optimal prior is identified after measuring the data and the notion of loss/risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (such computations are fast and do not suffer from the curse of dimensionality). The method is characterized by a parameter in [0,1] acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near 1, the method produces a posterior distribution concentrated around a maximum likelihood estimate (MLE) with tight but low confidence UQ estimates. When that parameter is near 0, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.