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# Quasi-Bayesian theory

We use the theory of convex sets of probability distributions, referred to as Quasi-Bayesian theory [Giron & Rios1980]. Motivation for the use of this theory in inferences, and a summary of the theory, is available in a companion technical report [Cozman1996]. Here we outline the chief ideas behind Quasi-Bayesian theory.

Imprecision in probability assessments can be due either to difficulties in eliciting information from experts, or to difficulties in processing or combining data [Walley1991]. This imprecision is modeled by convex sets of distributions, called credal sets. To simplify terminology, we use the term credal set only when it refers to a set of distributions containing more than one element. Convex sets of conditional distributions are used to represent conditional beliefs. Inference is performed by applying Bayes rule to each distribution in a prior credal set; the posterior credal set is the union of all posterior distributions. An introduction to technical aspects of Quasi-Bayesian theory, with a larger list of references, can be found at http://www.cs.cmu.edu/~fgcozman/qBayes.html.

Given a Quasi-Bayesian credal set K, a probability interval is induced for every event A:

p(A) = infp is in K p(A)

p(A) = supp is in K p(A).

A credal set always induces unique lower and upper bounds of probability, but arbitrary lower and upper probabilities do not induce a unique credal set.

Lower and upper expectations for a function u() are defined as:

E[xq] = minp Ep[u]

E[xq] = maxp Ep[u],

where Ep[u] is the usual expectation for function u() with probability distribution p().