  There are essentially three kinds of approaches to Uncertainty Quantification
(UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A)
is robust, it is unfavorable with respect to accuracy and data assimilation.
(B) requires a prior, it is generally brittle 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 risk
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 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 (which is fast and not subject to 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 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.
