A credal set is a set of probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s or, equivalently, a set of probability measure

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...

s. A credal set is often assumed or constructed to be a closed

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

 convex set

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

. It is intended to express uncertainty

Uncertainty

Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...

 or doubt about the probability model that should be used, or to convey the beliefs of a Bayesian

Bayesian probability

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

 agent about the possible states of the world.

Let denote a categorical variable, a probability mass function over , and a credal set over . If is convex, the credal set can be equivalently described by its extreme points . The expectation for a function of with respect to the credal set can be characterised only by its lower and upper bounds. For the lower we have . Notably, such a inference problem can be equivalently obtained by considering only the extreme points of the credal set.

It is easy to see that a credal set over a Boolean variable cannot have more than two vertices, while no bounds can be provided for credal sets over variables with three or more values.

See also

• imprecise probability
  Imprecise probability
  Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify...
  
  
• Dempster–Shafer theory
• p-boxes
  Probability box
  A probability box is a characterization of an uncertain number consisting of both aleatoric and epistemic uncertainties that is often used in risk analysis or quantitative uncertainty modeling where numerical calculations must be performed...
  
  
• robust Bayes
  Robust Bayes analysis
  Robust Bayes analysis, also called Bayesian sensitivity analysis, investigates the robustness of answers from a Bayesian analysis to uncertainty about the precise details of the analysis. An answer is robust if it does not depend sensitively on the assumptions and calculation inputs on which it is...
  
  
• upper and lower probabilities
  Upper and lower probabilities
  Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.Because...