Imprecise probability¶
Represent incomplete probabilistic commitment by a coherent set of admissible probability measures or equivalent lower and upper expectations, so conclusions expose a range and distinguish robust agreement from decisions that depend on an unresolved model choice.
Core Idea¶
Imprecise probability is a family of uncertainty models that permits a set of probability measures or noncoincident lower and upper probabilities or expectations instead of forcing one fully specified additive distribution; envelope models use \(\underline P(A)=\inf_{P\in\mathcal K}P(A)\) and \(\overline P(A)=\sup_{P\in\mathcal K}P(A)\) for a credal set \(\mathcal K\). partial commitments define a feasible family of precise laws, and lower and upper envelopes propagate the least and greatest compatible expectation through queries; a narrow interval records strong consensus, a wide interval records unresolved alternatives, and robust conclusions are those shared across the admissible family.
Scope of Application¶
Imprecise probability applies when the analyst can specify a measurable possibility space, a nonempty class of admissible probability measures or assessments, and events, gambles, or random quantities evaluated relative to that class and establish that the model intentionally leaves at least some probabilistic quantities non-singleton while satisfying its declared consistency or coherence conditions and preserving the set, envelope, or behavioral semantics used for updating and decision. The entry maps a family of formal models rather than declaring one universally correct interpretation or decision criterion; each application must state its consistency, updating, dependence, and action conventions.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because imprecise can be mistaken for inaccurate, numerically rounded, statistically estimated, or poorly measured, whereas the technical identity is a deliberate set-valued or lower-upper representation of probabilistic commitment. The disciplined statement is that the object counts as Imprecise probability exactly when the model intentionally leaves at least some probabilistic quantities non-singleton while satisfying its declared consistency or coherence conditions and preserving the set, envelope, or behavioral semantics used for updating and decision
Manages Complexity¶
The abstraction compresses credal sets, lower and upper probabilities, lower previsions, interval probabilities, probability boxes, belief functions, random sets, possibility measures, robust Bayesian classes, and sets of desirable gambles into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Abstract Reasoning¶
- Type the carrier. Establish a measurable possibility space, a nonempty class of admissible probability measures or assessments, and events, gambles, or random quantities evaluated relative to that class and reject examples from a different problem. 2. Lock the rule. Express that the model intentionally leaves at least some probabilistic quantities non-singleton while satisfying its declared consistency or coherence conditions and preserving the set, envelope, or behavioral semantics used for updating and decision independently of one notation or implementation.
Knowledge Transfer¶
Transfer within probability and decision theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from A closed convex credal set K contains every probability distribution satisfying declared linear constraints, and event A receives the interval formed by the minimum and maximum of P(A) over K. to A probability-box model bounds an uncertain cumulative distribution between lower and upper cumulative functions and propagates those bounds through a system model. demonstrates that continuity.
Relationships to Other Abstractions¶
Current abstraction Imprecise probability Domain-specific
Parents (1) — more general patterns this builds on
-
Imprecise probability is a kind of Probability Prime
The proposed strict upward parent is
prime:probability.
Hierarchy paths (2) — routes to 2 parentless roots
- Imprecise probability → Probability → Measure → Aggregation → Micro Macro Linkage
- Imprecise probability → Probability → Measure → Set and Membership
Neighborhood in Abstraction Space¶
Imprecise probability sits in a sparse region of the domain-specific corpus (60th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Imprecise Probability & Multiple Testing (5 abstractions)
Nearest neighbors
- Continuous-time stochastic process — 0.87
- Algebra of random variables — 0.87
- Sub-probability measure — 0.86
- Probability axioms — 0.86
- Cumulative distribution function — 0.86
Computed from structural-signature embeddings · 2026-09-08