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,[citation needed] 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 or Choquet (1953).
More precisely, in the work of these authors one considers in a power set, , a mass function satisfying the conditions
In turn, a mass is associated with two non-additive continuous measures called belief and plausibility defined as follows:
In the case where is infinite there can be such that there is no associated mass function. See p. 36 of Halpern (2003). Probability measures are a special case of belief functions in which the mass function assigns positive mass to singletons of the event space only.
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
The upper and lower probabilities are also related with probabilistic logic: see Gerla (1994).
Observe also that a necessity measure can be seen as a lower probability and a possibility measure can be seen as an upper probability.
and 25 Related for: Upper and lower probabilities information
Upperandlowerprobabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe...
evidence theory lowerandupperprobabilities, or interval probabilities belief functions possibility and necessity measures lowerandupper previsions comparative...
logic Uncertain inference Upperandlowerprobabilities James Franklin, The Science of Conjecture: Evidence andProbability before Pascal, 2001 The Johns...
interval cumulative probability distribution upperandlowerprobabilities credal set risk analysis uncertainty propagation probability bounds analysis Dempster–Shafer...
[citation needed] Imprecise probability Dempster–Shafer theory Probability box Robust Bayes analysis Upperandlowerprobabilities Levi, I. (1980). The Enterprise...
about both the center and the spread of the data. Knowing the lowerandupper quartile provides information on how big the spread is and if the dataset is...
a special case of an inference theory using upperandlowerprobabilities. Developing ideas of Fisher and of Pitman from 1938 to 1939, George A. Barnard...
count Unseen species problem Unsolved problems in statistics Upperandlowerprobabilities Upside potential ratio – finance Urn problem Ursell function...
begins with the digit 1 similarly has no natural density: the lower density is 1/9 and the upper density is 5/9. (See Benford's law.) Consider an equidistributed...
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different...
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b, and mode...
which the uniform distribution is a special case, and which is useful in estimating success probabilities. The four-parameter Beta distribution, a straight-forward...
two types, transition probabilitiesand emission probabilities (also known as output probabilities). The transition probabilities control the way the hidden...
the lower quartile), Q2 (the median), and Q3 (also called the upper quartile). The lower quartile corresponds with the 25th percentile and the upper quartile...
upper bunk and their corresponding edges in the lower bunk share the same probability. The probabilities assigned to the posts can be arbitrary. A random...
upper case Roman letters: X {\textstyle X} , Y {\textstyle Y} , etc. Particular realizations of a random variable are written in corresponding lower case...
To do so, one must calculate the probability that Pr(X = k) for all values k from 0 through n. (These probabilities should sum to a value close to one...
unknown distribution p will tend to assign higher probabilities q(xi) to the test events. Thus, they have lower perplexity because they are less surprised by...
(this upper bound is attained by A 5 {\displaystyle {\mathfrak {A}}_{5}} , the alternating group of degree 5). The set of commuting probabilities of finite...
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number...
limits are called the upper reference limit (URL) or upper limit of normal (ULN) and the lower reference limit (LRL) or lower limit of normal (LLN)....
Upper Bele, Lower Bele, Lower Kimbin (Kibin), andUpper Pyramid. Hupla, traditionally considered a separate language, is closer to Lower Grand Valley...
there is a 100γ% confidence that the parameter of interest is within a lowerandupper bound. A common misconception of confidence intervals is 100γ% of the...
distribution of the parameter and lCPOS is the lower bound of the credible interval of CPOS. The first criterion ensures that the probability of success is large...
Global Information
