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

• We study the consequences of the distinction between ontic and epistemic sets in statistical reasoning.
• We discuss the differences between three view of random sets.
• we lay bare examples where the different views matter: conditioning, variance and independence
• We show that interval regression may refer to distinct problems.
• We extend the discussion to fuzzy sets and random fuzzy sets.

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.