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.

Version: v3 · 2026-09-19 · History
Domain-specific #: 2047
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Probability and Decision Theory, Imprecise Uncertainty Models → Mathematics

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.

How would you explain it like I'm…

Somewhere-Between Chances

If someone asks how likely it is to rain, you might not know exactly. Instead of guessing one number, you honestly say 'somewhere between a little and a lot.' If your friends all agree, the gap is small; if they disagree, the gap stays big, and you keep it big instead of pretending.

Chance as a Range

Usually we describe chance with one exact number, like a 30% chance of rain. Imprecise probability lets you use a range instead, like 'between 20% and 50%,' when your knowledge only supports that much. You can think of it as keeping a whole group of possible chance-guesses that all fit what you know. A narrow range means the guesses agree; a wide range means you really have competing possibilities. A conclusion you can trust is one that comes out the same for every guess in the group.

Upper and Lower Probabilities

Imprecise probability is a way of modeling uncertainty when you can't justify one exact probability distribution. Instead, you use a set of distributions that all fit your partial knowledge, and report a lower and upper probability for each event: the smallest and largest values across the set. A narrow gap means the models agree; a wide gap means real unresolved alternatives. This is different from ordinary sampling error, from a single rough estimate, and from a 'confidence interval' — the width is not a coverage guarantee. It is also not the same as just saying 'I don't know'; the set must be coherent and its updating rules stated. Conclusions that hold across the whole set count as robust.

Imprecise probability is a family of uncertainty models that represents belief by a set of probability measures (a credal set) or by lower and upper probabilities/expectations, rather than a single additive distribution. For a credal set K, the lower probability of A is the infimum of P(A) over K and the upper is the supremum. Partial commitments define the feasible family of precise laws, and lower and upper envelopes propagate the least and greatest compatible expectations through queries. Interval width encodes disagreement among admissible models, not sampling error, fuzzy membership, or unstructured ignorance. Typed primitives include credal sets, lower previsions, capacities, p-boxes, and belief functions; each needs checks of coherence and lower/upper conjugacy, and explicit conditioning and independence assumptions. Errors include lower exceeding upper bounds, silently picking one distribution, treating width as confidence coverage, or deriving a decision rule from bounds alone. Uses include robust Bayesian and sensitivity analysis, propagating probability bounds, and handling scarce, conflicting, or partially elicited knowledge.

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.
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) — outgoing typed relations

Imprecise probability is a kind of Probability Prime

Every admissible member and every lower-upper envelope concern probability assignments or expectations; What makes it its own entry: deliberate coherent partial specification of probability through a family or lower-upper envelope, including explicit width and update semantics, rather than ordinary sampling uncertainty, one approximate probability, a fuzzy membership grade, or an unstructured admission of ignorance.

Hierarchy path (1) — routes to 1 parentless root

Imprecise probability → is a kind of Probability → is a kind of Measure → presupposes Set and Membership

Neighborhood in Abstraction Space¶

Imprecise probability sits in a moderately populated region (45^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Measure & Probability Foundations (42 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 · 2026-10-25