Fusion of qualitative beliefs

This chapter introduces the notion of qualitative belief assignment to model beliefs of human experts expressed in natural language (with linguistic labels). We show how qualitative beliefs can be efficiently combined using an extension of Dezert-Smarandache Theory (DSmT) of plausible and paradoxical quantitative reasoning to qualitative reasoning. We propose a new arithmetic on linguistic labels which allows a direct extension of classical or hybrid DSm fusion rules. An approximate qualitative PCR5 rule is also proposed jointly with a Qualitative Average Operator. We also show how crisp or interval mappings can be used to deal indirectly with linguistic labels. A very simple example is provided to illustrate our qualitative fusion rules. 10.1 A brief historic of previous works Since fifteen years qualitative methods for reasoning under uncertainty developed in Artificial Intelligence are attracting more and more people of Information Fusion community, specially those working in the development of modern multi-source1 systems for defense. Their aim is to propose solutions for processing and combining qualitative information to take into account efficiently information provided by human sources (or semi-intelligent expert systems) and usually expressed in natural language rather than direct quantitative information. George Polya was one of the first mathematicians to attempt a formal characterization of qualitative human reasoning in 1954 [26, 27], then followed by Lofti Zadeh’s works [40][45]. The interest of qualitative reasoning methods is to help in decision-making for situations in which the precise numerical methods are not appropriate (whenever the information/input are not directly expressed in numbers). Several formalisms for qualitative reasoning have been proposed as extensions on the frames of probability, possibility and/or evidence theories [2, 5, Where both computers, sensors and human experts are involved in the loop. 269 9, 12, 37, 39, 42, 45]. The limitations of numerical techniques are discussed in [22]. We browse here few main approaches. A detailed presentation of theses techniques can be found in [24]. 270 FUSION OF QUALITATIVE BELIEFS In [34], Wellman proposes a general characterization of qualitative probability to relax precision in representation and reasoning within the probabilistic framework. His basic idea was to develop Qualitative Probabilistic Networks (QPN) based on a Qualitative Probability Language (QPL) defined by a set of numerical underlying probability distributions. The major interest of QPL is to specify the partial rankings among degrees of belief rather than assessing their magnitudes on a cardinal scale. Such method cannot be considered as truly qualitative in our opinion, since it rather belongs to the family of imprecise probability [33] and probability bounds analysis (PBA) methods [11]. Some advances have been done by Darwiche in [5] for a symbolic generalization of Probability Theory; more precisely, Darwiche proposes a support (symbolic and/or numeric) structure which contains all information able to represent and conditionalize the state of belief. Darwiche shows that Probability Theory fits within his new support structure framework as several other theories, but Demspter-Shafer Theory doesn’t fit in. Based on Demspter-Shafer Theory [29] (DST), Wong and Lingras [38] propose a method for generating a (numerical) basic belief functions from preference relations between each pair of propositions be specified qualitatively. The algorithm proposed doesn’t provide however a unique solution and doesn’t check the consistency of qualitative preference relations. Bryson and al. [4, 16] propose a procedure called Qualitative Discriminant Procedure (QDP) that involves qualitative scoring, imprecise pairwise comparisons between pairs of propositions and an optimization algorithm to generate consistent imprecise quantitative belief function to combine. Very recently, Ben Yaglane in [1] has reformulated the problem of generation of quantitative (consistent) belief functions from qualitative preference relations as a more general optimization problem under additional non linear constraints in order to minimize different uncertainty measures (Bezdek’s entropy, Dubois & Prade non-specificity, etc). In [18, 19], Parsons proposes a qualitative Dempster-Shafer Theory, called Qualitative Evidence Theory (QET), by using techniques from qualitative reasoning [2]. Parsons’ idea is to use qualitative belief assignments (qba), denoted here qm(.) assumed to be only 0 or +, where + means some unknown value between 0 and 1. Parsons proposes, using operation tables, a very simple arithmetic for qualitative addition + and multiplication × operators. The combination of two (or more) qba’s then actually follows the classical conjunctive consensus operator based on his qualitative multiplication table. Because of impossibility of qualitative normalization, Parsons uses the un-normalized version of Dempster’s rule by committing a qualitative mass to the empty set following the open-world approach of Smets [32]. This approach cannot deal however with truly closed-world problems because there is no issue to transfer the conflicting qualitative mass or to normalize the qualitative belief assignments in the spirit of DST. An improved version of QET has been proposed [18] for using refined linguistic quantifiers as suggested by Dubois & Prade in [10]. The fusion of refined qualitative belief masses follows the un-normalized Dempster’s rule based on an underlying numerical interval arithmetic associated with linguistic quantifiers. Actually, this refined QTE fits directly within DSmT framework since it corresponds to imprecise (quantitative) DSmC fusion rule [6, 30]. From 1995, Parsons seems to have switched back to qualitative probabilistic reasoning [23] and started to develop Qualitative Probabilistic Reasoner (QPR). Recently, Parsons discussed about the flaw discovered in QPR and gave some issues with new open questions [25]. In Zadeh’s paradigm of computing with words (CW) [42][45] the combination of qualitative/vague information expressed in natural language is done essentially in three steps: 1) a 10.2. QUALITATIVE OPERATORS 271 translation of qualitative information into fuzzy membership functions, 2) a fuzzy combination of fuzzy membership functions; 3) a retranslation of fuzzy (quantitative) result into natural language. All these steps cannot be uniquely accomplished since they depend on the fuzzy operators chosen. A possible issue for the third step is proposed in [39]. In this chapter, we propose a simple arithmetic of linguistic labels which allows a direct extension of classical (quantitative) combination rules proposed in the DSmT framework into their qualitative counterpart. Qualitative beliefs assignments are well adapted for manipulated information expressed in natural language and usually reported by human expert or AI-based expert systems. In other words, we propose here a new method for computing directly with words (CW) and combining directly qualitative information Computing with words, more precisely computing with linguistic labels, is usually more vague, less precise than computing with numbers, but it is expected to offer a better robustness and flexibility for combining uncertain and conflicting human reports than computing with numbers because in most of cases human experts are less efficient to provide (and to justify) precise quantitative beliefs than qualitative beliefs. Before extending the quantitative DSmT-based combination rules to their qualitative counterparts, it will be necessary to define few but new important operators on linguistic labels and what is a qualitative belief assignment. Then we will show though simple examples how the combination of qualitative beliefs can be obtained in the DSmT framework. 10.2 Qualitative Operators We propose in this section a general arithmetic for computing with words (or linguistic labels). Computing with words (CW) and qualitative information is more vague, less precise than computing with numbers, but it offers the advantage of robustness if done correctly since : ” It would be a great mistake to suppose that vague knowledge must be false. On the contrary, a vague belief has a much better chance of being true than a precise one, because there are more possible facts that would verify it.” – Bertrand Russell [28]. So let’s consider a finite frame Θ = {θ1, . . . , θn} of n (exhaustive) elements θi, i = 1, 2, . . . , n, with an associated modelM(Θ) on Θ (either Shafer’s modelM0(Θ), free-DSm modelMf (Θ), or more general any Hybrid-DSm model [30]). A modelM(Θ) is defined by the set of integrity constraints on elements of Θ (if any); Shafer’s model M0(Θ) assumes all elements of Θ truly exclusive, while free-DSm model Mf (Θ) assumes no exclusivity constraints between elements of the frame Θ. Let’s define a finite set of linguistic labels L̃ = {L1, L2, . . . , Lm} where m ≥ 2 is an integer. L̃ is endowed with a total order relationship ≺, so that L1 ≺ L2 ≺ . . . ≺ Lm. To work on a close linguistic set under linguistic addition and multiplication operators, we extends L̃ with two extreme values L0 and Lm+1 where L0 corresponds to the minimal qualitative value and Lm+1 corresponds to the maximal qualitative value, in such a way that L0 ≺ L1 ≺ L2 ≺ . . . ≺ Lm ≺ Lm+1 where ≺ means inferior to, or less (in quality) than, or smaller (in quality) than, etc. hence a relation of order from a qualitative point of view. But if we make a correspondence between qualitative labels and quantitative values on the scale [0, 1], then Lmin = L0 would correspond 272 FUSION OF QUALITATIVE BELIEFS to the numerical value 0, while Lmax = Lm+1 would correspond to the numerical value 1, and each Li would belong to [0, 1], i. e. Lmin = L0 < L1 < L2 < . . . < Lm < Lm+1 = Lmax From now on, we work on extended ordered set L of qualitative values L = {L0, L̃, Lm+1} = {L0, L1, L2, . . . , Lm, Lm+1} The qualitative addition and multiplication operators are respectively defined in the following way: • Addition : Li + Lj = { Li+j, if i+ j < m+ 1, Lm+1, if i+ j ≥ m+ 1. (10.1) • Multiplication : Li × Lj = Lmin{i,j} (10.2) These two operators are well-defined, commutative, associative, and unitary. Addition of labels is a unitary operation since L0 = Lmin is the unitary element, i.e. Li+L0 = L0+Li = Li+0 = Li for all 0 ≤ i ≤ m+ 1. Multiplication of labels is also a unitary operation since Lm+1 = Lmax is the unitary element, i.e. Li×Lm+1 = Lm+1×Li = Lmin{i,m+1} = Li for 0 ≤ i ≤ m+1. L0 is the unit element for addition, while Lm+1 is the unit element for multiplication. L is closed under + and ×. The mathematical structure formed by (L,+,×) is a commutative bisemigroup with different unitary elements for each operation. We recall that a bisemigroup is a set S endowed with two associative binary operations such that S is closed under both operations. If L is not an exhaustive set of qualitative labels, then other labels may exist in between the initial ones, so we can work with labels and numbers since a refinement of L is possible. When mapping from L to crisp numbers or intervals, L0 = 0 and Lm+1 = 1, while 0 < Li < 1, for all i, as crisp numbers, or Li included in [0, 1] as intervals/subsets. For example, L1, L2, L3 and L4 may represent the following qualitative values: L1 , very poor, L2 , poor, L3 , good and L4 , very good where , symbol means ”by definition”. We think it is better to define the multiplication × of Li×Lj by Lmin{i,j} because multiplying two numbers a and b in [0, 1] one gets a result which is less than each of them, the product is not bigger than both of them as Bolanos et al. did in [3] by approximating Li × Lj = Li+j > max{Li, Lj}. While for the addition it is the opposite: adding two numbers in the interval [0, 1] the sum should be bigger than both of them, not smaller as in [3] case where Li + Lj = min{Li, Lj} < max{Li, Lj}. 10.2.1 Qualitative Belief Assignment We define a qualitative belief assignment (qba), and we call it qualitative belief mass or q-mass for short, a mapping function qm(.) : G 7→ L 10.2. QUALITATIVE OPERATORS 273 where GΘ corresponds the space of propositions generated with ∩ and ∪ operators and elements of Θ taking into account the integrity constraints of the model. For example if Shafer’s model is chosen for Θ, then GΘ is nothing but the classical power set 2Θ [29], whereas if free DSm model is adopted GΘ will correspond to Dedekind’s lattice (hyper-power set) DΘ [30]. Note that in this qualitative framework, there is no way to define normalized qm(.), but qualitative quasi-normalization is still possible as seen further. Using the qualitative operations defined previously we can easily extend the combination rules from quantitative to qualitative. In the sequel we will consider s ≥ 2 qualitative belief assignments qm1(.), . . . , qms(.) defined over the same space GΘ and provided by s independent sources S1, . . . , Ss of evidence. Important note: The addition and multiplication operators used in all qualitative fusion formulas in next sections correspond to qualitative addition and qualitative multiplication operators defined in (10.1) and (10.2) and must not be confused with classical addition and multiplication operators for numbers. 10.2.2 Qualitative Conjunctive Rule (qCR) The qualitative Conjunctive Rule (qCR) of s ≥ 2 sources is defined similarly to the quantitative conjunctive consensus rule, i.e. qmqCR(X) = ∑ X1,...,Xs∈G X1∩...∩Xs=X s ∏