A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, where it is used as a way of measuring the expected utility of an uncertain event. It is applied specifically to membership functions and capacities. In imprecise probability theory, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone lower probability, or the upper expectation induced by a 2-alternating upper probability.
Contents
- Definition
- Properties
- Monotonicity
- Positive homogeneity
- Comonotone additivity
- Subadditivity
- Superadditivity
- Alternative Representation
- Applications
- References
Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the Ellsberg paradox and the Allais paradox.
Definition
The following notation is used:
Assume that
Then the Choquet integral of
where the integrals on the right-hand side are the usual Riemann integral (the integrands are integrable because they are monotone in
Properties
In general the Choquet integral does not satisfy additivity. More specifically, if
for some functions
The Choquet integral does satisfy the following properties.
Monotonicity
If
Positive homogeneity
For all
Comonotone additivity
If
then
Subadditivity
If
Superadditivity
If
Alternative Representation
Let
where
Applications
The Choquet integral was applied in image processing, video processing and computer vision.