Probabilis

Probabilis is a theoretical framework in probability theory and decision making that generalizes the classical probability measure by using interval-valued likelihoods. For every event A in a measurable space, probabilis assigns a lower probabilis L(A) and an upper probabilis U(A), with 0 ≤ L(A) ≤ U(A) ≤ 1. The interval [L(A), U(A)] represents the degree of belief about A given partial information, evidence, or ambiguity. If L(A) = U(A) = p for all A, the framework reduces to a standard probability measure.

A compatible pair (L, U) satisfies basic coherence constraints: L(∅) = 0, U(Ω) = 1, and L(A) ≤ U(A) for all A. Monotonicity holds: if A ⊆ B then L(A) ≤ L(B) and U(A) ≤ U(B). In typical constructions, L and U also obey subadditivity and superadditivity relations that ensure consistency across unions and intersections. Specific models implement these relations to reflect how evidence combines or conflicts.

Probabilis encompasses classical probability as a special case while also covering imprecise or interval probabilities. It relates to and draws on ideas from Dempster–Shafer theory, belief and plausibility, and coherent lower previsions, providing a unifying language for expressing both precise beliefs and partial information.

Applications of probabilis appear in risk assessment, reliability engineering, finance, and artificial intelligence decision systems where information is incomplete or conflicting. Computational approaches often rely on bounds, approximations, or sampling to derive useful inferences about complex events.

See also: probability theory, interval probability, imprecise probability, Dempster–Shafer theory, robust Bayesian.