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 frequentist statistics disallows metaprobabilities, frequentists have had to propose new solutions. Cedric Smith and Arthur Dempster each developed a theory of upper and lower probabilities. Glenn Shafer developed Dempster's theory further, and it is now known as Dempster-Shafer theory (see also Choquet53).More precisely, in these authors one considers in a power set, P(S),!, a "mass" function m : P(S) ightarrow R satisfying the conditions

:m(varnothing) = 0 ,,,,,,! ; ,,,,,, sum_{A in P(X)} m(A) = 1. ,!

In turn, a mass is associated with two non-additive continuous measures called belief and plausibility defined as follows:

:operatorname{bel}(A) = sum_{B mid B subseteq A} m(B),,,,;,,,,operatorname{pl}(A) = sum_{B mid B cap A e varnothing} m(B)

A different notion of upper and lower probabilities is obtained by the "lower and upper envelopes" obtained from a class "C" of probability distributions by setting :operatorname{env_1}(A) = inf_{p in C} p(A),,,,;,,,,operatorname{env_2}(A) = sup_{p in C} p(A)

The upper and lower probabilities are also related with probabilistic logic (see Gerla94).

Observe also that a necessity measure can be seen as a lower probability and a possibility measure can be seen as an upper probability.

References

* G. Gerla, Inferences in Probability Logic, "Artificial Intelligence" 70(1–2):33–52, 1994.

* J. Y. Halpern and R. Fagin, Two views of belief: Belief as generalized probability and belief as evidence. "Artificial Intelligence", 54:275-317, 1992.

* P. J. Huber, "Robust Statistics". Wiley, New York, 1980.

* Saffiotti, A., A Belief-Function Logic, in "Procs of the 10h AAAI Conference", San Jose, CA 642-647, 1992.

* Choquet, G., Theory of Capacities, "Annales de l'Institut Fourier" 5, 131-295, 1953.

* Shafer, G., "A Mathematical Theory of Evidence", (Princeton University Press, Princeton), 1976.

* P. Walley and T. L. Fine, Towards a frequentist theory of upper and lower probability. "Annals of Statistics", 10(3):741-761, 1982.

ee also

* Possibility theory
* Probability theory
* Fuzzy measure theory


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium

  • Method of conditional probabilities — In mathematics and computer science, the probabilistic method is used to prove the existence of mathematical objects with desired combinatorial properties. The proofs are probabilistic they work by showing that a random object, chosen from some… …   Wikipedia

  • Entropy in thermodynamics and information theory — There are close parallels between the mathematical expressions for the thermodynamic entropy, usually denoted by S , of a physical system in the statistical thermodynamics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s; and the …   Wikipedia

  • Exact sciences (The) in Hellenistic times: texts and issues — The exact sciences in Hellenistic times: Texts and issues1 Alan C.Bowen Modern scholars often rely on the history of Greco Latin science2 as a backdrop and support for interpreting past philosophical thought. Their warrant is the practice… …   History of philosophy

  • Imprecise probability — The notion of Imprecise probability is used as a generic term to cover all mathematical models which measure chance or uncertainty without sharp numerical probabilities. It includes both qualitative (comparative probability, partial preference… …   Wikipedia

  • Dempster–Shafer theory — Prof Arthur P. Dempster at the workshop on Belief Function Theory (Brest 1 april 2010). The Dempster–Shafer theory (DST) is a mathematical theory of evidence.[1] It allows …   Wikipedia

  • Probabilistic logic — The aim of a probabilistic logic (or probability logic) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. The result is a richer and more expressive formalism with a… …   Wikipedia

  • Credal set — A credal set is a set of probability distributions[1] or, equivalently, a set of probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability… …   Wikipedia

  • Possibility theory — is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D.… …   Wikipedia

  • List of mathematics articles (U) — NOTOC U U duality U quadratic distribution U statistic UCT Mathematics Competition Ugly duckling theorem Ulam numbers Ulam spiral Ultraconnected space Ultrafilter Ultrafinitism Ultrahyperbolic wave equation Ultralimit Ultrametric space… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”