Probabilistic Merging Operators

The merging/fusion of belief/data collections in propositional logic form is a topic that has received due attention within the domains of database and AI research. A distinction can be made between two types of scenarios to which the process of merging can be applied. In the first type, the collections represent preferences, such as the voting choices of a group of people, that need to be aggregated so as to give a consistent result that in some way best represents the collective judgement of the group. In the second type, the collections represent factual data that is to be aggregated with an aim of obtaining a result that maximises factual correctness. After introducing a general framework for belief merging via some prominent literature on the topic, this paper then introduces and considers a method for belief merging with the second type of scenario in mind. Its suitability is corroborated by demonstrating how it can be seen as a special case of a merging procedure that combines aggregation of probabilities and maximisation of expected truthlikeness.

2012

Belief merging aims at extracting a coherent and informative view from a set of belief bases. A first requirement for belief merging operators is to obey basic rationality conditions. Another expected property is to preserve as much information as possible from the input bases. In this paper, we show how new merging operators, called compositional operators, can be defined from existing ones. Such operators aim at offering a higher discriminative power than the merging operators on which they are based, without leading to a complexity shift or losing rationality postulates. We identify some sufficient conditions for ensuring that rationality is fully preserved by composition.

A Meeting of the Minds, Proceedings LORI Beijing, 2007

Proceedings of the Twentieth International Conference on Principles of Knowledge Representation and Reasoning

In standard propositional belief merging, one implicit assumption is that all sources have exactly the same importance. But there are many situations where the sources have different importance/reliability/expertise that have to be taken into account in the merging process. In this work we study the problem of weighted merging operators, which aimed to take these weights into account in a sensible way. We give a syntactical characterization of these operators, and then we state a representation theorem in terms of plausibility preorders on interpretations. We also propose a general method to build weighted distance-based merging operators, and provide some concrete examples, using two different weight functions.

2017

Symbolic and Quantitative …, 2007

When merging belief functions, Dempster rule of combination is justified only when information sources can be considered as independent. When this is not the case, one must find out a cautious merging rule that adds a minimal amount of information to the inputs. Such a rule is said to follow the principle of minimal commitment. Some conditions it should comply with are studied. A cautious merging rule based on maximizing expected cardinality of the resulting belief function is proposed. It recovers the minimum operation when specialized to possibility distributions. This form of the minimal commitment principle is discussed, in particular its discriminating power and its justification when some conflict is present between the belief functions.

Information Fusion, 2010

The problem of fusing beliefs in the Dempster-Shafer belief theory has attracted considerable attention over the last two decades. The classical Dempster's Rule has often been criticised, and many alternative rules for belief fusion have been proposed in the literature. We show that it is crucial to consider the nature of the situation to be modelled and to select the appropriate fusion operator as a function thereof. In this paper we present the cumulative and averaging fusion rules for belief functions, which represent generalisations of the subjective logic cumulative and averaging fusion operators for opinions respectively. The generalised operators are applicable to the combination of general basic belief assignments (bbas). These rules, which can be directly derived from classical statistical theory, produce results that correspond well with human intuition.

Springer eBooks, 2014

On setting of general information (i.e.without probability) we define, by axiomatic way, general misinformation of an event. We give a class of measures of misinformation, solving a sistem of functional equations, given by the properties of the misinformation. Then, we propose some aggregation operators of these general misinformation.

We consider the problem of merging several belief bases in the presence of integrity con- straints. We propose a logical characterization of operators having a majority behaviour or a consensual one. Then we give a representation theorem in terms of pre-orders on interpre- tations. We show the close connection between belief revision and merging operators and we show that our proposal extends the pure merging case (i.e. without integrity constraints) we study in a previous work. Finally we show that Liberatore and Schaerf commutative revision operators can be seen as a special case of merging.

Different methods have been proposed for merging multiple and potentially conflicting informations. Sum-based operators offer a natural method for merging commensurable prioritized belief bases. Their popularity is due to the fact that they satisfy the majority property and they adopt a non cautious attitude in deriving plausible conclusions. This paper analyses the sum-based merging operator when sources to merge are incommensurable, namely they do not share the same meaning of uncertainty scales. We first show that the obtained merging operator can be equivalently characterized either in terms of an infinite set of compatible scales, or by a well-known Pareto ordering on a set of models. We then study different families of compatible scales useful for merging process. This paper also provides a postulates-based analysis of our merging operators.

2010

Belief merging is often described as the process of defining a base which best represents the beliefs of a group of agents (a profile of belief bases). The resulting base can be viewed as a synthesis of the input profile. In this paper another view of what belief merging aims at is considered: the epistemic view. Under this view the purpose of belief merging is to best approximate the true state of the world. We point out a generalization of Condorcet's Jury Theorem from the belief merging perspective. Roughly, we show that if the beliefs of sufficiently many reliable agents are merged then in the limit the true state of the world is identified. We introduce a new postulate suited to the truth tracking issue. We identify some merging operators from the literature which satisfy it and other operators which do not.

The problem of aggregating pieces of propositional information coming from several agents has given rise to an intense research activity. Two distinct theories have emerged. On the one hand, belief merging has been considered in AI as an extension of belief revision. On the other hand, judgment aggregation has been developed in political philosophy and social choice theory. Judgment aggre-gation focusses on some specific issues (represented as formulas and gathered into an agenda) on which each agent has a judgment, and aims at defining a collective judgment set (or a set of collective judgment sets). Belief merging considers each source of information (the belief base of each agent) as a whole, and aims at defining the beliefs of the group without considering an agenda. In this work the relationships between the two theories are investigated both in the general case and in the fully informed case when the agenda is complete (i.e. it contains all the possible interpretations). Thoug...

Workshop on the Theory on Belief …, 2010

When merging belief functions, Dempster rule of combination is justified only when information sources can be considered as independent and reliable. When dependencies are ill-known, it is usual to require the combination rule to be idempotent, as it ensures a cautious behaviour in the face of dependent sources. There are different strategies to find such rules for belief functions. The strategy considered here consists in relying on idempotent rules used in a more specific frameworks and to study its extension to belief functions. We study two possible extensions of the minimum rule of possibility theory to belief functions. We first investigate under which conditions it can be extended to general contour functions.We then further investigate the combination rule that maximises the expected cardinality of the resulting random set.

"ABSTRACT Within the framework of discrete probabilistic uncertain reasoning a large literature exists justifying the maximum entropy inference process, ME, as being optimal in the context of a single agent whose subjective probabilistic knowledge base is consistent. In [9] Paris and Vencovska, extending the work of Johnson and Shore [6], completely characterised the ME inference process by an attractive set of axioms which an inference process should satisfy, thus providing a quite different justification for ME from that of the more traditional possible worlds or information theoretic arguments whose origins go back to nineteenth century statistical mechanics as in [8] or [5]. More recently the second author in [10] and [11] extended the Paris- Vencovska axiomatic approach to inference processes to the context of several agents whose subjective probabilistic knowledge bases, while in- dividually consistent, may be collectively inconsistent. In particular he defines a "social entropy process", SEP, which is a natural extension of the single agent ME. However, while SEP is known to possess many attractive properties, these are almost certainly insufficient to uniquely characterise SEP. It is therefore of particular interest to study those Paris-Vencovska principles valid for ME whose immediate generalisations to the multiagent case are not satisfied by SEP. One of these principles is the Irrelevant Information Principle, a principle which very few inference processes satisfy even in single agent context. In this paper we will inves- tigate whether SEP can satisfy an interesting modified generalisation of this principle."

In this work, we de ne some non-prioritized merge operators, that is, operators for the consistent union of belief bases. We de ne some postulates for several kinds of merge operator and we give different constructions: trivial merge, partial meet merge and kernel merge. For some constructions we provide representation theorems linking construction with a set of postulates. Finally, we propose that the formulated operators can be used in some multi-agent systems.