tag:blogger.com,1999:blog-3722233.post3168893514408307020..comments2021-01-13T04:02:17.369-06:00Comments on Computational Complexity: Imagining Imaginary ProbabilitiesLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-3722233.post-4626037341980993652012-10-04T21:26:07.042-05:002012-10-04T21:26:07.042-05:00This reminds me of the idea of a "half coin.&...This reminds me of the idea of a "half coin." A half coin is one that if you flip it twice you get exactly what a single coin toss looks like. This turns out to have a fairly precise definition--a sensiable mean, variance, moment generating function, central limit theorem, etc. What it doesn't have is probabilities that live between zero and one!<br /><br />Dean Fosterhttps://www.blogger.com/profile/13906505132477512644noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-13239418674774671042012-10-02T00:29:03.156-05:002012-10-02T00:29:03.156-05:00If the "interpretation" is that probabil...If the "interpretation" is that probabilities p are not constant, but represent expectations over some function y of unwritten parameters t (e.g. a time series), then at least it's consistent to have both E[y_t = H] = p_t = 1/2, and E[y_t = H and y_{t+1} = T] = p_t (1 - p_{t+1}) = 1/2.<br /><br />But at the moment I can't think of an interpretation for an imaginary solution to the arbitrary equation of two elements of the series that isn't nonsense, without introducing more arbitrary structure over the parameters. (Like in QM.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-43089017546054940532012-09-18T23:52:57.420-05:002012-09-18T23:52:57.420-05:00With some effort you could probably extend this to...With some effort you could probably extend this to complex surreal numbers.<br />http://en.m.wikipedia.org/wiki/Surreal_numberLumierehttps://www.blogger.com/profile/08829954049453695191noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-79327580315700857222012-09-17T20:12:51.103-05:002012-09-17T20:12:51.103-05:00Why just have probability? Complex measure theory ...Why just have probability? Complex measure theory lets you have imaginary masses, distances, volumes, etc. Everything you need for a surreal existence.Chrishttps://www.blogger.com/profile/02873949286995651782noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-72142274719465335912012-09-17T16:07:23.843-05:002012-09-17T16:07:23.843-05:00Last year I heard a nice talk by Persi Diaconis on...Last year I heard a nice talk by Persi Diaconis on a probability theory with negative numbers and the analogues of standard theorems (such as the central limit theorem) one can prove in this theory.<br /><br />An online version seems to be here:<br />http://vimeo.com/29445127Anonymoushttps://www.blogger.com/profile/14199670490757426241noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-12775894350916004342012-09-17T12:14:20.240-05:002012-09-17T12:14:20.240-05:00FYI: http://www.scottaaronson.com/democritus/lec9....FYI: http://www.scottaaronson.com/democritus/lec9.htmlPeter Mazsahttp://mazsa.comnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-69739328864527038322012-09-17T12:05:30.387-05:002012-09-17T12:05:30.387-05:00You were on your way to re-discovering quantum inf...You were on your way to re-discovering quantum information until you decided to insist on the solution being "non-quantum". :)aram harrowhttps://www.blogger.com/profile/01272118188252697149noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-45465577451910533322012-09-17T09:40:19.463-05:002012-09-17T09:40:19.463-05:00If I recall correctly, imaginary probabilities fig...If I recall correctly, imaginary probabilities figured in Isaac Asimov's Foundation Trilogy.stuhttps://www.blogger.com/profile/05190631846507740664noreply@blogger.com