Probability bounds analysis: Difference between revisions

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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|date=June 2023}} 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>)