Conditioning and inference in intelligent systems

Abstract

We maintain that among the research trends concerning the various aspects and methodologies for the management of partial and revisable information in automated reasoning and giving particular emphasis to conditioning and inference, conditional events and conditional probability (in a coherent -in the sense of de Finetti -framework) play a central role: we will review some of our and related results, showing that this approach is able to encompass many of other existing numerical and symbolic methods dealing with the treatment of partial knowledge and uncertainty in Artificial Intelligence.

The role of coherence for handling probabilistic evaluations and independence

“…We refer to a definition of conditional event discussed, e.g., in [10,13,15] (which differs, e.g., from that given in [21,25]), where a conditional event, denoted by EjH , is an ordered pair ðE; H Þ of events, with H 6 ¼ ;, such that its truth-value T ðEjH Þ (which generalizes the concept of indicator concerning an event E) can be represented as a three-valued random variable…”

“…We refer to [10,13] for a wide discussion about the truthvalue of conditional events and for a natural interpretation of T ðEjH Þ in terms of betting scheme, that may help in clarifying its meaning. Moreover, in [13] the authors show that taking two commutative, associative and increasing operations È and from R þ Â R þ to R þ and suitably defining corresponding operations among the set T of three-valued random variables, the ''result'' is a random variable, but it does not in general belong to T ; then considering only those elements of T 2 such that the range of each operation is in T , the induced condition on tðEjH Þ can be taken as ''natural'' axioms for conditional measures.…”

“…Moreover, in [13] the authors show that taking two commutative, associative and increasing operations È and from R þ Â R þ to R þ and suitably defining corresponding operations among the set T of three-valued random variables, the ''result'' is a random variable, but it does not in general belong to T ; then considering only those elements of T 2 such that the range of each operation is in T , the induced condition on tðEjH Þ can be taken as ''natural'' axioms for conditional measures. Then different conditional measures can be obtained by particular choices of the two operations È and : for example choosing ordinary sum and product, the authors in [10] get conditional probability; while choosing maximum and minimum conditional possibilities are obtained [2]. We recall now the axioms of conditional probabilities in the sense of de Finetti [18], Krauss [26], Dubins [20].…”

Nonmonotonic probabilistic reasoning under variable-strength inheritance with overriding

“…Another important approach to probabilistic reasoning with conditional constraints is based on the coherence principle of de Finetti and generalizations of it [7,11,12,13,14,27,28,29,54], or on similar principles that have been adopted for lower and upper probabilities [51,57]. The main tasks in this framework are checking the consistency of a probabilistic assessment, and the propagation of a given assessment to further conditional events.…”

Probabilistic Logic under Coherence, Model-Theoretic Probabilistic Logic, and Default Reasoning

“…In such cases, a general approach is obtained by using (conditional and/or unconditional) probabilistic constraints, based on the coherence principle of de Finetti and suitable generalizations of it [5,8,9,10,11,20,21,22,37], or on similar principles that have been adopted for lower and upper probabilities [36,41]. Two important aspects in dealing with uncertainty are: (i) checking the consistency of a probabilistic assessment, and (ii) the propagation of a given assessment to further uncertain quantities.…”