Several counterparts of Bayesian networks based on different paradigms have been proposed in evidence theory. Nevertheless, none of them is completely satisfactory. In this paper we will present a new one, based on a recently introduced concept of conditional independence. We define a conditioning rule for variables, and the relationship between conditional independence and irrelevance is studied with the aim of constructing a Bayesian-network-like model. Then, through a simple example, we will show a problem appearing in this model caused by the use of a conditioning rule. We will also show that this problem can be avoided if undirected or compositional models are used instead.
We present an alternative approach to decision-making in the framework of possibility theory, based on the idea of decision-making under uncertainty. We utilize the fact, that any possibility distribution can be viewed as an upper envelope of a set of probability distributions to which well-known minimax principle is applicable. Finally, we recall also an alternative to the minimax rule, so-called local minimax principle. Local minimax principle not only allows sequential construction of decision function, but also appears to play an important role exactly in the framework of possibility theory due to its sensitivity. Furthermore, the optimality of a decision function is easily verifiable.
A possibilistic marginal problem is introduced in a way analogous to probabilistic framework, to address the question of whether or not a common extension exists for a given set of marginal distributions. Similarities and differences between possibilistic and probabilistic marginal problems will be demonstrated, concerning necessary condition and sets of all solutions. The operators of composition will be recalled and we will show how to use them for finding a T-product extension. Finally, a necessary and sufficient condition for the existence of a solution will be presented.
In contrast to most other approaches used to represent multidimensional probability distributions, which are based on graphical Markov modelling (i.e. dependence structure of distributions is represented by graphs), the described method is rather procedural. Here, we describe a process by which a multidimensional distribution can be composed from a “generating sequence” - a sequence of low-dimensional distributions. The main advantage of this approach is that the same apparatus based on operators of composition can be applied for description of both probabilistic and possibilistic models.