Ambiguity and robust statistics

Since the seminal work of Gilboa and Schmeidler [28, p. 142] a relation between decision making under ambiguity and robust Bayesian statistics has been hinted at, and indeed immediate similarities are quite evident. At the same time, a formal treatment of this topic and a complete characterization of the relation between the two approaches is still missing. The object of this paper is to …ll the gap, that is, relating ambiguity (also called Knightian uncertainty or model uncertainty) to prior uncertainty. Ambiguity refers to the case in which a decision maker does not know the probability distribution governing the stochastic nature of the problem he is facing. This uncertainty is captured by using nonadditive probabilities –capacities –or sets of probability measures over the space of states of the world.1 Prior uncertainty, in a parametric statistical model fP g 2 , refers to uncertainty about the prior on . This is a classical problem in robust Bayesian statistics, where uncertainty is again represented by capacities or sets of priors over the space of parameters.2 Clearly, prior uncertainty can be reduced (literally) to ambiguity. Loosely, if is a (prior) capacity on the space of parameters, then