Content deleted Content added
GreenC bot (talk | contribs) Reformat 1 archive link. Wayback Medic 2.5 |
m cleanup, fixing lint errors |
||
Line 4:
This [[upper and lower bounds|bounding]] approach permits analysts to make calculations without requiring overly precise assumptions about parameter values, dependence among variables, or even distribution shape. Probability bounds analysis is essentially a combination of the methods of standard [[interval analysis]] and classical [[probability theory]]. Probability bounds analysis gives the same answer as interval analysis does when only range information is available. It also gives the same answers as [[Monte Carlo simulation]] does when information is abundant enough to precisely specify input distributions and their dependencies. Thus, it is a generalization of both interval analysis and probability theory.
The diverse methods comprising probability bounds analysis provide algorithms to evaluate mathematical expressions when there is uncertainty about the input values, their dependencies, or even the form of mathematical expression itself. The calculations yield results that are guaranteed to enclose all possible distributions of the output variable if the input [[probability box|p-boxes]] were also sure to enclose their respective distributions. In some cases, a calculated p-box will also be best-possible in the sense that the bounds could be no tighter without excluding some of the possible distributions.
P-boxes are usually merely bounds on possible distributions. The bounds often also enclose distributions that are not themselves possible. For instance, the set of probability distributions that could result from adding random values without the independence assumption from two (precise) distributions is generally a proper [[subset]] of all the distributions enclosed by the p-box computed for the sum. That is, there are distributions within the output p-box that could not arise under any dependence between the two input distributions. The output p-box will, however, always contain all distributions that are possible, so long as the input p-boxes were sure to enclose their respective underlying distributions. This property often suffices for use in [[Probabilistic risk assessment|risk analysis]] and other fields requiring calculations under uncertainty.
==History of bounding probability==
The idea of bounding probability has a very long tradition throughout the history of probability theory. Indeed, in 1854 [[George Boole]] used the notion of interval bounds on probability in his [[The Laws of Thought]].<ref name="BOOLE1854">{{cite book|url= https://www.gutenberg.org/ebooks/15114 |last=Boole |first=George |title=An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities |publisher=Walton and Maberly |year=1854 |___location=London}}</ref><ref name=Hailperin86>{{cite book |last=Hailperin |first=Theodore |title=Boole's Logic and Probability |publisher=North-Holland |year=1986 |___location=Amsterdam |isbn=978-0-444-11037-4 }}</ref> Also dating from the latter half of the 19th century, the [[Chebyshev inequality|inequality]] attributed to [[Chebyshev]] described bounds on a distribution when only the mean and variance of the variable are known, and the related [[Markov inequality|inequality]] attributed to [[Andrey Markov|Markov]] found bounds on a positive variable when only the mean is known. [[Henry E. Kyburg, Jr.|Kyburg]]<ref name="kyburg99">Kyburg, H.E., Jr. (1999). [http://www.sipta.org/documentation/interval_prob/kyburg.pdf Interval valued probabilities]. SIPTA Documention on Imprecise Probability.</ref> reviewed the history of interval probabilities and traced the development of the critical ideas through the 20th century, including the important notion of incomparable probabilities favored by [[John Maynard Keynes|Keynes]].
Of particular note is [[Maurice René Fréchet|Fréchet]]'s derivation in the 1930s of bounds on calculations involving total probabilities without dependence assumptions. Bounding probabilities has continued to the present day (e.g., Walley's theory of [[imprecise probability]].<ref name="WALLEY1991">{{cite book|url= https://archive.org/details/statisticalreaso0000wall |last=Walley |first=Peter |title=Statistical Reasoning with Imprecise Probabilities |url-access=registration |publisher=Chapman and Hall |year=1991 |___location=London |isbn=978-0-412-28660-5 }}</ref>)
The methods of probability bounds analysis that could be routinely used in
Line 126 ⟶ 93:
==Magnitude comparisons==
The probability that an uncertain number represented by a p-box ''D'' is less than zero is the interval Pr(''D'' < 0) = [<u>''F''</u>
:''A'' < ''B'' = Pr(''A'' − ''B'' < 0),
:''A'' > ''B'' = Pr(''B'' − ''A'' < 0),
| |||