The primary difference between precise and imprecise probability theories lies in the allowance for imprecision, or a gap between upper and lower expectations (also called previsions) of bounded real functions. This gap generates a set of probability distributions or measures. As a result, in imprecise probabilities, the notion of independence on joint spaces is not unique; for example, notions of unknown interaction, epistemic irrelevance/independence and strong independence have been proposed in the literature. After introducing the three concepts of independence, various algorithms are proposed to calculate, through the different definitions of independence, both prevision and conditional probability bounds generated by marginal distributions over finite joint spaces,. All algorithms are designed to accommodate two different types of constraints that define the sets of marginal distributions: previsions bounds or extreme distributions. Algorithms are applied to simple examples that show the role of the different quantities introduced and the equivalence of the two types of constraints. It is shown that, in epistemic irrelevance/independence, re-writing algorithms in terms of joint distributions turn quadratic optimization problems into linear ones.
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