Agregating Quantitative Possibilistic Networks

The problem of merging multiple-source uncertain information is a crucial issue in many applications. This paper proposes an analysis of possibilistic merging operators where uncertain information is encoded by means of product-based (or quantitative) possibilistic networks. We first show that the product-based merging of possibilistic networks having the same DAG structures can be easily achieved in a polynomial time. We then propose solutions to merge possibilistic networks having different structures with the help of additionnal variables. Introduction The problem of combining pieces of information issued from different sources can be encountered in various fields of applications such as databases, multi-agent systems, expert opinion pooling, etc. Several works have been recently achieved to fuse propositional or weighted logical knowledge bases issued from different sources (Baral et al. 1992),(Cholvy 1998), (Konieczny and Pérez 1998), (Lin 1996), (Lin and Mendelzon 1998), (Benferhat et al. 1997). This paper addresses the problem of fusion of uncertain pieces of information represented by possibilistic networks. Possibilistic networks (Fonck 1992; Borgelt et al. 1998; Gebhardt and Kruse 1997) are important tools proposed for an efficient representation and analysis of uncertain information. Their success is due to their simplicity and their capacity of representing and handling independence relationships which are important for an efficient management of uncertain pieces of information. Possibilistic networks are directed acyclic graphs (DAG), where each node encodes a variable and every edge represents a relationship between two variables. Uncertainties are expressed by means of conditional possibility distributions for each node in the context of its parents. In possibility theory, there are two kinds of possibilistic causal networks depending if possibilistic conditioning is Copyright c © 2006, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. based on the minimum or on the product operator. In the rest of this paper, we only consider product-based conditioning. In this case, possibilistic networks are called quatitative (or product-based) possibilistic networks. The rest of this paper is organised as follows. Next section gives a brief background on possibility theory and quantitative possibilistic networks. Section 3 recalls the conjunctive combination mode on possibility distributions. Section 4 discusses the fusion of possibistic networks having same graphical structures. Section 5 deals with fusion of possibilistic networks having different structures but the union of their DAGs is free of cycles. Section 6 proposes a general approach for merging any set of possibilistic networks. Section 7 concludes the paper. Basics of possibility theory Let V = {A1, A2, ..., AN} be a set of variables. We denote by DA = {a1, .., an} the domain associated with the variable A. By a we denote any instance of A. Ω = ×Ai∈V DAi denotes the universe of discourse, which is the Cartesian product of all variable domains in V . Each element ω ∈ Ω is called a state of Ω. In the following, we only give a brief recalling on possibility theory, for more details see (Dubois and Prade 1988). Possibility distributions and possibility measures A possibility distribution π is a mapping from Ω to the interval [0, 1]. It represents a state of knowledge about a set of possible situations distinguishing what is plausible from what is less plausible. Given a possibility distribution π defined on the universe of discourse Ω, we can define a mapping grading the possibility measure of an event φ ⊆ Ω by Π(φ) = maxω∈φ π(ω). A possibility distribution π is said to be normalized, if h(π) = maxω π(ω) = 1. In a possibilistic setting, conditioning consists in modifying our initial knowledge, encoded by a possibility distribution π, by the arrival of a new sure piece of information φ ⊆ Ω. There are different definitions of condition-