Based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence), we adopt a probabilistic approach to uncertainty based on conditional probability bounds. Our notion of g-coherence is equivalent to the "avoiding uniform loss" property for lower and upper probabilities (a la Walley). Moreover, given a g-coherent imprecise assessment by our algorithms we can correct it obtaining the associated coherent assessment (in the sense of Walley and Williams). As is well known, the problem of checking g-coherence and/or propagating conditional probability bounds has, in general, an exponential complexity. Two notions which may be helpful to reduce computational effort are those of non relevant gain and basic set. Exploiting them, our algorithms can use linear systems with reduced sets of variables and/or linear constraints. In this paper we give some insights on the notions of non relevant gain and basic set. We consider several families with three conditional events, obtaining some results characterizing g-coherence in such cases. We also give some more general results.
Keywords. uncertain knowledge, coherence, g-coherence, imprecise probabilities, conditional probability bounds, lower and upper probabilities, non relevant gains, basic sets.
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Authors addresses:
Veronica Biazzo
Citta Universitaria
Viale A. Doria 6
95152 Catania
Italy
Angelo Gilio
Dipartimento di Metodi e Modelli Matematici
Via A. Scarpa 16, 00161 Roma (Italy)
Giuseppe Sanfilippo
Citta Universitaria
Viale A. Doria 6
95152 Catania
Italy
E-mail addresses:
| Veronica Biazzo | vbiazzo@dmi.unict.it |
| Angelo Gilio | gilio@dmmm.uniroma1.it |
| Giuseppe Sanfilippo | gsanfilippo@dmi.unict.it |