Robust Bayesianism: Relation to Evidence Theory

Several, apparently incomparable, approaches exist for uncertainty management. Uncertainty management is a broad area applied in many different fields, where information about some underlying, not directly observable, truth–the state of the world–is sought from a set of observations that are more or less reliable. These observations can be, for example, measurements with random and/or systematic errors, sensor readings, or reports submitted by observers. In order that conclusions about the conditions of interest be possible, there must be some assumptions made on how the observations relate to the underlying state about which information is sought. Most such assumptions are numerical in nature, giving a measure that indicates how plausible different underlying states are. Such measures can usually be normalized so that the end result looks very much like a probability distribution over the possible states of the world, or over sets of possible world states. However, uncertainty management and information fusion is often concerned with complex technical, social or biological systems that are incompletely understood, and it would be naive to think that the relationship between observation and state can be completely captured. At the same time, such systems must have at least some approximate ways to relate observation with state in order to make uncertainty management at all possible. It has been a goal in research to encompass all aspects of uncertainty management in a single framework. Attaining this goal should make the topic teachable in undergraduate and graduate engineering curricula and facilitate engineering applications development. We propose here that robust Bayesian analysis is such a framework. The Dempster-Shafer or evidence theory originated within Bayesian statistical analysis [19], but when developed by Shafer [51] took the concept of belief assignment rather than probability distribution as primitive. The assumption being that bodies of evidence–beliefs about the possible worlds of interest–can be taken as primitives rather than sampling functions and priors. Although this idea has had considerable popularity, it is inherently dangerous since it seems to move application away from foundational justification. When the connection to Bayes’ method and Dempster’s application model is broken, it is no longer necessary to use the Dempster combination rule, and evidence theory abounds with proposals on how bodies of evidence should be interpreted and combined, as a rule with convincing but disparate argumentation. But there seems not to exist other bases for obtaining bodies of evidence than likelihoods and priors, and therefore an analysis of a hypothetical Bayesian obtainment of bodies of evidence can bring light to problems in evidence theory. Particularly, a body of evidence represented by a DS-structure has an interpretation as a set of possible probability distributions, and combining or aggregating two such structures can be done in robust