Risk, ambiguity, and misspecification: Decision theory, robust control, and statistics

What are "deep uncertainties" and how should their presence influence prudent decisions? To address these questions, we bring ideas from robust control theory into statistical decision theory. Decision theory has its origins in axiomatic formulations by von Neumann and Morgenstern, Wald, and Savage. After Savage, decision theorists constructed axioms that formalize a notion of ambiguity aversion. Meanwhile, control theorists constructed decision rules that are robust to some model misspecifications. We reinterpret axiomatic foundations of decision theories to express ambiguity about a prior over a family of models along with concerns about misspecifications of the corresponding likelihood functions.