The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle.
As a simple example for n=2, one can think of darts being thrown at a board, with their landing spots in the plane distributed according to a 2-variable normal distribution centered at the origin. (This is a reasonable assumption for any given darts player, with different players being described by different normal distributions.) If we now consider a circle and a rectangle in the plane, both centered at the origin, then the proportion of the darts landing in the intersection of both shapes is no less than the product of the proportions of the darts landing in each shape. This can also be formulated in terms of conditional probabilities: if you're informed that your last dart hit the rectangle, then this information will increase your estimate of the probability that the dart hit the circle.
History
A special case of the inequality was conjectured in 1955;[1] further development was given by Olive Jean Dunn in 1958.[2][3] The general case was stated in 1972, also as a conjecture.[4] The case of dimension n=2 was proved in 1977[5] and certain special cases of higher dimension have also been proven in subsequent years.[6]
The general case of the inequality remained open until 2014, when Thomas Royen, a retired German statistician, proved it using relatively elementary tools.[7] In fact, Royen generalized the conjecture and proved it for multivariate gamma distributions. The proof did not gain attention when it was published in 2014, due to Royen's relative anonymity and the fact that the proof was published in a predatory journal.[2][8] Another reason was a history of false proofs (by others) and many failed attempts to prove the conjecture, causing skepticism among mathematicians in the field.[2]
The conjecture, and its solution, came to public attention in 2017, when other mathematicians described Royen's proof in a mainstream publication[9] and popular media reported on the story.[2][10][11]
References
^Dunnett, C. W.; Sobel, M. (1955). "Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t -distribution". Biometrika. 42 (1–2): 258–267. doi:10.1093/biomet/42.1-2.258. ISSN 0006-3444.
^ abcdWolchover, Natalie (March 28, 2017). "A Long-Sought Proof, Found and Almost Lost". QUANTA magazine. Retrieved April 4, 2017.
^Schechtman, G.; Schlumprecht, Th.; Zinn, J. (January 1998). "On the Gaussian measure of the intersection". The Annals of Probability. 26 (1): 346–357. doi:10.1214/aop/1022855422. ISSN 0091-1798.
^Das Gupta, S.; Eaton, M. L.; Olkin, I.; Perlman, M.; Savage, L. J.; Sobel, M. (1972). Inequalities on the probability content of convex regions for elliptically contoured distributions. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. II: Probability theory. Berkeley, California: Univ. California Press. pp. 241–265.
^Pitt, Loren D. (1977). "A Gaussian Correlation Inequality for Symmetric Convex Sets". The Annals of Probability. 5 (3): 470–474. doi:10.1214/aop/1176995808.
^Harge, Gilles (1999). "A Particular Case of Correlation Inequality for the Gaussian Measure" (PDF). The Annals of Probability. 27 (4): 1939–1951. doi:10.1214/aop/1022874822. S2CID 118789169.
^Royen, Thomas (November 2014). "A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions". Far East Journal of Theoretical Statistics. 48 (2): 139–145. arXiv:1408.1028.
^"Pushpa Publishing House". www.pphmj.com. Retrieved 4 July 2017.
^Latała, R.; Matlak, D. (2017). "Royen's Proof of the Gaussian Correlation Inequality". Geometric Aspects of Functional Analysis; Israel Seminar (GAFA) 2014–2016. Lecture Notes in Mathematics. 2169: 265–275. arXiv:1512.08776. doi:10.1007/978-3-319-45282-1_17. ISBN978-3-319-45281-4. S2CID 36992343.
^Farand, Chloe (2017-04-03). "Retired man solves one of hardest maths problems in the world and no one notices". The Independent. Retrieved 2017-04-04.
^Dambeck, Holger (2017-04-04). "Erfolg mit 67 Jahren: Der Wunderopa der Mathematik". SPIEGEL ONLINE. Retrieved 2017-04-04.
External links
George Lowther, The Gaussian Correlation Conjecture, "Almost Sure"
March 09, 2023
gaussian, correlation, inequality, formerly, known, gaussian, correlation, conjecture, mathematical, theorem, fields, mathematical, statistics, convex, geometry, gaussian, correlation, inequality, states, that, probability, hitting, both, circle, rectangle, wi. The Gaussian correlation inequality GCI formerly known as the Gaussian correlation conjecture GCC is a mathematical theorem in the fields of mathematical statistics and convex geometry The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle Contents 1 The statement 2 History 3 References 4 External linksThe statement EditThe Gaussian correlation inequality states Let m displaystyle mu be an n dimensional Gaussian probability measure on R n displaystyle mathbb R n i e m displaystyle mu a multivariate normal distribution centered at the origin Then for all convex sets E F R n displaystyle E F subset mathbb R n that are symmetric about the origin m E F m E m F displaystyle mu E cap F geq mu E cdot mu F As a simple example for n 2 one can think of darts being thrown at a board with their landing spots in the plane distributed according to a 2 variable normal distribution centered at the origin This is a reasonable assumption for any given darts player with different players being described by different normal distributions If we now consider a circle and a rectangle in the plane both centered at the origin then the proportion of the darts landing in the intersection of both shapes is no less than the product of the proportions of the darts landing in each shape This can also be formulated in terms of conditional probabilities if you re informed that your last dart hit the rectangle then this information will increase your estimate of the probability that the dart hit the circle History EditA special case of the inequality was conjectured in 1955 1 further development was given by Olive Jean Dunn in 1958 2 3 The general case was stated in 1972 also as a conjecture 4 The case of dimension n 2 was proved in 1977 5 and certain special cases of higher dimension have also been proven in subsequent years 6 The general case of the inequality remained open until 2014 when Thomas Royen a retired German statistician proved it using relatively elementary tools 7 In fact Royen generalized the conjecture and proved it for multivariate gamma distributions The proof did not gain attention when it was published in 2014 due to Royen s relative anonymity and the fact that the proof was published in a predatory journal 2 8 Another reason was a history of false proofs by others and many failed attempts to prove the conjecture causing skepticism among mathematicians in the field 2 The conjecture and its solution came to public attention in 2017 when other mathematicians described Royen s proof in a mainstream publication 9 and popular media reported on the story 2 10 11 References Edit Dunnett C W Sobel M 1955 Approximations to the probability integral and certain percentage points of a multivariate analogue of Student s t distribution Biometrika 42 1 2 258 267 doi 10 1093 biomet 42 1 2 258 ISSN 0006 3444 a b c d Wolchover Natalie March 28 2017 A Long Sought Proof Found and Almost Lost QUANTA magazine Retrieved April 4 2017 Schechtman G Schlumprecht Th Zinn J January 1998 On the Gaussian measure of the intersection The Annals of Probability 26 1 346 357 doi 10 1214 aop 1022855422 ISSN 0091 1798 Das Gupta S Eaton M L Olkin I Perlman M Savage L J Sobel M 1972 Inequalities on the probability content of convex regions for elliptically contoured distributions Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Vol II Probability theory Berkeley California Univ California Press pp 241 265 Pitt Loren D 1977 A Gaussian Correlation Inequality for Symmetric Convex Sets The Annals of Probability 5 3 470 474 doi 10 1214 aop 1176995808 Harge Gilles 1999 A Particular Case of Correlation Inequality for the Gaussian Measure PDF The Annals of Probability 27 4 1939 1951 doi 10 1214 aop 1022874822 S2CID 118789169 Royen Thomas November 2014 A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions Far East Journal of Theoretical Statistics 48 2 139 145 arXiv 1408 1028 Pushpa Publishing House www pphmj com Retrieved 4 July 2017 Latala R Matlak D 2017 Royen s Proof of the Gaussian Correlation Inequality Geometric Aspects of Functional Analysis Israel Seminar GAFA 2014 2016 Lecture Notes in Mathematics 2169 265 275 arXiv 1512 08776 doi 10 1007 978 3 319 45282 1 17 ISBN 978 3 319 45281 4 S2CID 36992343 Farand Chloe 2017 04 03 Retired man solves one of hardest maths problems in the world and no one notices The Independent Retrieved 2017 04 04 Dambeck Holger 2017 04 04 Erfolg mit 67 Jahren Der Wunderopa der Mathematik SPIEGEL ONLINE Retrieved 2017 04 04 External links EditGeorge Lowther The Gaussian Correlation Conjecture Almost Sure Retrieved from https en wikipedia org w index php title Gaussian correlation inequality amp oldid 1137981616, wikipedia, wiki, book, books, library,
