Risk, Ambiguity, and Misspecification: Decision Theory, Robust Control, and Statistics

What are “deep uncertainties” and how should their presence alter prudent courses of action? To help answer these questions, we bring ideas from robust control theory into statistical decision theory. Decision theory in economics has its origins in axiomatic formulations by von Neumann and Morgenstern as well as the statisticians Wald and Savage. Since Savage’s fundamental work, economists have provided alternative axioms that formalize a notion of ambiguity aversion. Meanwhile, control theorists created another way to construct decision rules that are robust to potential model misspecifications. We reinterpret axiomatic foundations of some modern decision theories to include ambiguity about a prior to put on a family of models simultaneously with concerns about misspecifications of the corresponding likelihood functions. By building on ideas from dynamic programming, our representations have recursive structures that preserve dynamic consistency.