Locally strong coherence and inference with lower–upper probabilities A. Capotorti, L. Galli, B. Vantaggi Abstract We introduce an operational way to reduce the spatial complexity in inference processes based on con- ditional lower–upper probabilities assessments. To reach such goal we must suitably exploit zero probabilities taking account of logical conditions characterizing locally strong coherence. We actually re-formulate for conditional lower–upper probabilities the notion of locally strong co- herence already introduced for conditional precise prob- abilities. Thanks to the characterization, we avoid to build all atoms, so that several real problems become feasible. In fact, the real complexity problem is connected to the number of atoms. Since for an inferential process with lower–upper probabilities several sequences of constraints must be fulfilled, our simplification can have either a ‘‘global’’ or a ‘‘partial’’ effect, being applicable to all or just to some sequences. The whole procedure has been implemented by XLisp-Stat language. A comparison with other approaches will be done by an example. Keywords Lower-upper probabilities, Inference, Locally strong coherence 1 Introduction Our paper deals with the problem of finding a sound and efficient algorithm for an inferential process based on partial conditional lower–upper probability assessments. First of all, let us discuss two main features of this task: partial assessments and conditional lower–upper proba- bilities. The adoption of partial assessments is due to their generality and wide applicability. Its roots are in de Finetti’s ideas (see for instance [12, 13]) and nowadays has found a wide application (as an incomplete, but relevant, reference see [5, 6, 8, 9, 15]). The peculiarity of incom- pleteness (i.e. the domain of the valuation is not neces- sarily a structured set, like an algebra or a r-algebra) distinguishes this theory from the usual ones (based on a measure-theoretic approach), where a complete (on structured domains) initial evaluation is required. Sometimes the completeness hypothesis is not realistic in applications, hence partial assessments can better repre- sent the actual problem. On the contrary, such generality has the drawback that the assessment must be coherent with a specified numer- ical framework and this must be operationally checked. In this paper we will adopt an approach based on an interval valued conditional probability assessment to represent uncertainty and, in particular, we refer to a class of conditional probabilities whose lower (and upper) envelope coincides with the extreme values of the assess- ment. Moreover, we deal with the inference problem, i.e. to extend a conditional lower–upper assessment to new events, and we are able to propose a unified algorithm for both purposes: checking coherence and making inference. Our procedure, as suggested in [7] and developed in [2–4] for precise assessments, allows to split the domain into different ‘‘layers’’ and reduce the space complexity. Peculiarity of such method is the ‘‘admissibility’’ and the ‘‘usefulness’’ of conditioning events with zero probabilities (since layers are determined by them). This will lead to a procedure for checking the coherence and making inference based on the solutions of sequences of linear programming problems. The notion of ‘‘locally strong coherence’’ can be inter- preted as a condition detecting suitable subfamilies ‘‘not influencing’’ coherence, so that they can be ‘‘thrown away’’: this technique drastically reduces dimensions of matrices and number of sequences (of linear systems). Obviously, what we already did for precise assessments [2–4] is useful also for lower–upper probabilities because the coherence condition in this case is weaker. Anyhow such notion can be re-formulated for the new framework. Such generalization leads also to a partial simplification of some of the sequences. Its usefulness will be shown by a simple example, in comparison with other approaches. Both properties (global and partial locally strong coherence) have been characterized by logical conditions, so that an automatic procedure (implemented by XLisp-Stat language) based on them has been easily realized. 2 Coherence and extension of lower probability assessments One of the main feature of the (so-called) subjective ap- proach to uncertainty (based on the concept of coherence) is the powerful opportunity given to the decision maker to assess his ‘‘degree of belief’’ only on the propositions Focus Soft Computing 7 (2003) 280–287 Ó Springer-Verlag 2003 DOI 10.1007/s00500-002-0214-6 280 A. Capotorti (&), L. Galli 1 Dipartimento di Matematica e Informatica, Via Vanvitelli 1, 06123 Perugia, Italy E-mail: capot@dipmat.unipg.it B. Vantaggi Dipartimento di Metodi e Modelli Matematici, Via Scarpa 16, 00161 Roma, Italy 1 The contribution of L. Galli has been essentially addressed to some aspects of the algorithm’s implementation.