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The Epistemic View of Belief Merging: Can We Track the Truth?

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The Epistemic View of Belief Merging: Can We Track the Truth? The Epistemic View of Belief Merging: Can We Track the Truth? Patricia Everaere 1 and Sebastien´ Konieczny2 and Pierre Marquis2 Abstract. Belief merging is often described as the process of defin- In this paper we introduce a new point of view about belief merg- ing a base which best represents the beliefs of a group of agents (a ing, that goes beyond the usual synthesis view: the epistemic view. profile of belief bases). The resulting base can be viewed as a syn- Synthesis View: Under the synthesis view, belief merging aims at thesis of the input profile. In this paper another view of what belief characterizing a base which best represents the beliefs of the input merging aims at is considered: the epistemic view. Under this view profile. This is the view adopted in previous belief merging works. the purpose of belief merging is to best approximate the true state of Epistemic View: Under the epistemic view, the purpose of a belief the world. We point out a generalization of Condorcet’s Jury Theo- merging process is to best approximate the true state of the world. rem from the belief merging perspective. Roughly, we show that if the beliefs of sufficiently many reliable agents are merged then in In the general case, no agent has a perfect view of the real world, her the limit the true state of the world is identified. We introduce a new beliefs are pervaded with uncertainty: An agent typically does not know which one of the models of her postulate suited to the truth tracking issue. We identify some merg- • ing operators from the literature which satisfy it and other operators base represents the true state of the world, She is not even ensured that the true state of the world is really which do not. • among the models of her base.3 Belief merging under the epistemic view can be considered as a way 1 Introduction to circumvent such an uncertainty at the group level, by tracking the In many areas of computer science, including distributed databases true state of the world. Interestingly, the truth tracking issue discrimi- and multi-agent systems, one needs to synthesize pieces of informa- nates belief merging from goal merging. Indeed, the concept of truth tion issued from several sources. What makes this problem difficult tracking is meaningless when goals are considered: there is no no- is, among other things, that the information sources typically contra- tion of ”true goal” which would be analogous to the true state of the dict each other. When the available pieces of information are beliefs world in the goal merging setting. represented in propositional logic, this problem is called (proposi- The problem of truth tracking has been studied for centuries in so- tional) belief merging. Many different belief merging operators have cial choice and in political science, in order to justify the foundations been pointed out so far. In this paper we focus on the purely logical of decisions made by jury trials. The main theoretical result here is case, i.e., we assume that belief bases are sets of propositional formu- Condorcet’s Jury Theorem [7]. This theorem states that if a jury is lae (see e.g. [2, 20, 12, 11, 9]). Other operators have been provided composed of reliable and independent individuals who aim at deter- in more general settings, such as weighted logics (possibilistic logic mining the right answer to a yes/no question, then the probability that or settings based on ordinal conditional functions) [3, 16, 4], which the decision made by the jury is the right one tends to 1 as the size of prove useful when more information are available (especially, when the jury tends to infinity. all the pieces of belief are not equally certain). In these more general In this paper we formalize the truth tracking issue from a belief settings, the merging problem becomes close to preorder (preference) merging perspective. We show that some belief merging operators aggregation, as studied in social choice theory [1, 19]. can be used to identify the true state of the world by considering suf- Logical properties of merging operators have been investigated in ficiently many reliable, homogeneous and independent agents. To be several works [20, 14, 12]. In [12] a set of logical properties have more precise, we present a generalization of Condorcet’s Jury Theo- been put forward to characterize the family of IC (Integrity Con- rem under uncertainty (i.e., when each base may have several mod- straints) merging operators. IC merging operators have been advo- els). We introduce a Truth Tracking (TT) postulate, and we point cated to be suited to both belief merging and goal merging. Even if IC operators satisfying it and other IC operators which do not. This it might look strange at a first glance that very different concepts, shows that (TT) is independent of the (conjunction of the) IC postu- such as goals and beliefs, can be handled in the same way with re- lates. We also provide experimental results in order to investigate the spect to aggregation, the adequacy of IC merging operators to propo- convergence speed of truth tracking for the belief merging operator dD ,⌃ sitional merging (whatever goals or beliefs are to be merged) has not ∆ . In most cases the number of agents to be considered for en- been challenged so far. This makes sense since in both cases merging suring that the merged base identifies the true state of the world with aims at synthesizing the information represented in the given profile high probability is not so large. of propositional bases. 3 If one supposes that the agent is ensured that the true state of the world is a model of her belief base, then one talks about “knowledge” this 1 LIFL-CNRS, USTL, France, email: [email protected] assumption is the only difference between belief and knowledge − and 2 CRIL-CNRS, Universite´ d’Artois, France, email: [email protected], marq- knowledge merging is not so interesting, since the only sensible knowledge− [email protected] merging operator is conjunction. The rest of the paper is organized as follows. In the next section, 3 Condorcet’s Jury Theorem and Extensions we give some formal preliminaries. Then we recall Condorcet’s Jury Theorem, and some of its generalizations. In the third section we We first consider a profile Ec of n agents where each agent i votes point out new generalizations of Condorcet’s Jury Theorem suited to for an alternative, let us say a state of the world !i . Among the ? 2W the belief merging perspective. Then we introduce the (TT) postulate possible states of the world is the true one ! . and we show that IC postulates and (TT) are logically independent. The hypotheses used in Condorcet’s Jury Theorem are that agents Finally we present the experimental results we obtained and we dis- are both independent and reliable. Since several notions of relia- cuss them. For space reasons, we report the proof of the main result bility will be considered in the following, we call the first one R1- (Theorem 3), and omit the other ones. reliability: The R1-reliability pi of an agent i is the probability that i gives • ? 2 Preliminaries the true state of the world, i.e., pi = P ([Ki]= ! ). { } We consider a propositional language defined from a finite set of An agent i is R1-reliable if her R1-reliability is strictly greater L • propositional variables and the usual connectives. than 0.5. (R1) P For any subset c of , c denotes the number of elements of c. The majority rule simply returns as result the interpretation which P | | An interpretation (or state of the world) is a total function from receives a strict majority of votes. Formally, let us first define the P to 0, 1 . The set of all interpretations is noted . The true state of notion of score s(!) of a state of the world ! with respect to a profile { } ? W the world is noted ! . An interpretation ! is a model of a formula Ec of complete bases: s(!)= !i Ec s.t. !i = ! . |{ 2 }| φ if and only if it makes it true in the usual truth functional 2L way. [φ] denotes the set of models of formula φ, i.e., [φ]= ! Definition 1 (Majority) Given a profile Ec of n complete bases, the { 2 ! = φ . majority rule m is defined as: m(Ec)=! if s(!) >n/2. W| | } A base K denotes the set of beliefs of an agent, it is a finite set of propositional formulae, interpreted conjunctively. We identify K We are now ready to recall Condorcet’s Jury Theorem. In this the- with the conjunction of its elements. Basically, a base K represents orem, only two alternatives are considered so that each agent votes a set [K] of states of the world. for one of them: A profile E denotes the beliefs from the group of n agents that ? are involved in the merging process. In the following, agents ex- Theorem 1 ([7]) Consider a set = !,! consisting of two W { } press sometimes a single world, only. In this case, a profile Ec is possible states of the world and a profile Ec of complete bases from a vector of complete bases. In order to avoid heavy notations, we a set of n independent and R1-reliable agents sharing the same R1- assimilate each complete base with its model and write such pro- reliability. The probability that the majority rule on this profile re- ? files as Ec = !1,...,!n .
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