Probability bounds analysis: Difference between revisions

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{{Short description|Mathematical method of risk analysis}}

{{Use dmy dates|date=October 2022}}

'''Probability bounds analysis''' ('''PBA''') is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random variables and other quantities through mathematical expressions. For instance, it computes sure bounds on the distribution of a sum, product, or more complex function, given only sure bounds on the distributions of the inputs. Such bounds are called [[probability box]]es, and constrain [[cumulative distribution function|cumulative probability distributions]] (rather than [[probability density function|densities]] or [[probability mass function|mass functions]]).

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==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). [httphttps://www.sipta.org/documentation/interval_prob/kyburgkyburgnew.pdf Interval valued probabilities]. {{deadlink}} SIPTA Documentation 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>)

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:<math>\left \{ \overline{F}, \underline{F}, m, v, \mathbf{F} \right \},</math>

where <math>\overline{F}, </math> and <math>\underline{F} \in \mathbb{D}</math>, <math>m,</math> and <math>v</math> are real intervals, and <math>\mathbf{F} \subset \mathbb{D}.</math>. This quintuple denotes the set of distribution functions <math>F \in \mathbf{F} \subset \mathbb{D}</math> such that:

:<math>\begin{align}