Statistical reasoning with set-valued information: Ontic vs. epistemic views

Abstract

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. In information processing tasks, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading, a set represents an object of interest and its elements are subparts of the object, forming a composite description. In the disjunctive reading, a set contains mutually exclusive elements and refers to the representation of incomplete knowledge. It does not model an actual object or quantity, but partial information about an underlying object or a precise quantity. This distinction between what we call ontic vs. epistemic sets remains valid for fuzzy sets, whose membership functions, in the disjunctive reading are possibility distributions, over deterministic or random values. This paper examines the impact of this distinction in statistics. We show its importance because there is a risk of misusing basic notions and tools, such as conditioning, distance between sets, variance, regression, etc. when data are set-valued. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.

Ranking of fuzzy intervals seen through the imprecise probabilistic lens

“…We consider that these fuzzy sets model our uncertainty about values, that is we adopt the epistemic view [11]. There are at least two possible interpretations about these values.…”

Belief Revision and the EM Algorithm

“…However, we have shown cases of incomplete information management where this method does not seem to work properly. In future works we shall propose a more systematic analysis of situations when the EM algorithm stops at the first iteration under the partition assumption, and explore alternative ways of posing the problem of maximum likelihood estimation under incomplete overlapping data [2,9,10,7]. Another issue is to investigate the cogency of the fake sample found by the EM algorithm viewed as an imputation method [15,14].…”

Fuzzy data analysis and classification

“…• The methodologies based on the notion of fuzzy information system, introduced by Okuda et al (1978), who consider the available information from a classical random experiment associated with a real-valued random variable to be fuzzy [i.e., they consider an epistemic viewpoint in accordance with the distinction made by Couso and Dubois (2014)] and assume that this available information constitutes a fuzzy partition (in Ruspini's sense Ruspini 1970) of the sample space of the variable, and probabilities are based on Zadeh's probabilistic definition of fuzzy events (Zadeh 1968). Some data analysis developments using this model can be seen, for instance, in Gil et al (1988), Gil (1992) and, more recently, Denoeux (2011).…”