Lower probability models for uncertainty and nondeterministic processes

An extension to the class of conventional numerical probability models for nondeterministic phenomena has been identified by Dempster and Shafer in the class of belief functions. We were originally stimulated by this work, but have since come to believe that the bewildering diversity of uncertainty and chance phenomena cannot be encompassed within either the conventional theory of probability, its relatively minor modifications (e.g., not requiring countable additivity), or the theory of belief functions. In consequence, we have been examining the properties of, and prospects for, the generalization of belief functions that is known as upper and lower, or interval-valued, probability. After commenting on what we deem to be problematic elements of common personalist/subjectivist/Bayesian positions that employ either finitely or countably additive probability to represent strength of belief and that are intended to be normative for rational behavior, we sketch some of the ways in which the set of lower envelopes, a subset of the set of lower probabilities that contains the belief functions, enables us to preserve the core of Bayesian reasoning while admitting a more realistic (e.g., in its reduced insistence upon an underlying precision in our beliefs) class of probability-like models. Particular advantages of lower envelopes are identified in the area of the aggregation of beliefs.

The focus of our own research is in the area of objective probabilistic reasoning about time series generated by physical or other empirical (e.g., societal) processes. As it is not the province of a general mathematical methodology such as probability theory to a priori rule out of existence empirical phenomena, we are concerned by the contraint imposed by conventional probability theory that an empirical process of bounded random variables that is believed to have a time- invariant generating mechanism must then exhibot long-run stable time averages. We have shown that lower probability models that allow for unstable time averages can only lie in the class of undominated lower probabilities, a subset of lower probability models disjoint from the lower envelopes and having the weakest relationship to conventional probability measures. Our research has been devoted to exploring and developing the theory of undominated lower probabilities so that it can be applied to model and understand nondeterministic phenomena, and we have also been interested in identifying actual physical processes (e.g., flicker noises) that exhibit behavior requiring such novel models.