Bounds on Previsions and Conditional Probabilities on Joint Finite Spaces Under the Assumption of Independence in Imprecise Probability

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.

ON THE CHECKING OF G-COHERENCE OF CONDITIONAL PROBABILITY BOUNDS

We illustrate an approach to uncertain knowledge based on lower conditional probability bounds. We exploit the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence), which is equivalent to the "avoiding uniform loss" property introduced by Walley for lower and upper probabilities. Based on the additive structure of random gains, we define suitable notions of non relevant gains and of basic sets of variables. Exploiting them, the linear systems in our algorithms can work with reduced sets of variables and/or constraints. In this paper, we illustrate the notions of non relevant gain and of basic set by examining several cases of imprecise assessments defined on families with three conditional events. We adopt a geometrical approach, obtaining some necessary and sufficient conditions for g-coherence. We also propose two algorithms which provide new strategies for reducing the number of constraints and for deciding g-coherence. In this way, we try to overcome the computational difficulties which arise when linear systems become intractable. Finally, we illustrate our methods by giving some examples.