Ignorance and Imprecise Probabilities

Decisions or inferences sometimes have to made in situations where substantive information about aspects of the problem are either lacking or conflicting. This is often handled by constructing a non-informative prior by appealing to principles such as indifference, maximum entropy, invariance, or maximizing missing information. Unfortunately these priors and the resulting posteriors may depend on arbitrary choices about the sample space. The priors may be improper resulting in incoherence as well as possibly leading to inadmissible decisions despite a proper posterior (e.g., the usual estimator in the James-Stein problem). And on a more foundational level, there are reasonable arguments against using a precise prior distribution to represent ignorance about a problem. This talk discusses a methodological and philosophical alternative called Imprecise Probabilities which differs from the traditional Bayesian approach by modeling probabilities—or more generally previsions (expectations)—with intervals rather than single real numbers. In particular I will focus on the theory formalized by Walley (1991) also known as Coherent Lower Previsions. Starting from axioms of rational betting behavior, generalized notions of coherence and avoidance of sure loss are applied in a context where fair prices for gambles may not be determinable, resulting in gaps between the maximum buying prices (lower previsions) and the minimum selling prices (upper previsions). Ignorance is then represented as these gaps instead of attempting to specify a non-informative prior distribution.