Probability theory over noncommutative ring? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:32:33Z http://mathoverflow.net/feeds/question/52669 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52669/probability-theory-over-noncommutative-ring Probability theory over noncommutative ring? Tegiri Nenashi 2011-01-20T18:46:21Z 2013-02-25T18:22:00Z <p>Observation: <em>Entropy</em> is a metric over some non commutative ring. Indeed, if we exponentiate the standard entropy definition </p> <p>$\displaystyle H(X) = -\sum_{x \in \mathcal{X}} p(x) \ln p(x).$</p> <p>we'll get</p> <p>$\displaystyle exp(H(X)) = -\prod_{x \in \mathcal{X}} p(x)^{p(x)}.$</p> <p>The later is just a sum of squares if we interpret multiplication as summation and power as multiplication. This triggers all kind of questions, the most important one: can summation of probabilities be demoted and, possibly, entirely excluded from the theory?</p> http://mathoverflow.net/questions/52669/probability-theory-over-noncommutative-ring/121513#121513 Answer by Tegiri Nenashi for Probability theory over noncommutative ring? Tegiri Nenashi 2013-02-11T18:21:14Z 2013-02-11T18:21:14Z <p>Section 2.5 of David Ellerman's <a href="http://www.ellerman.org/Davids-Stuff/Maths/Counting-Dits-reprint.pdf" rel="nofollow">"Counting distinctions: on the conceptual foundations of Shannonâ€™s information theory"</a> overviews the history of the sum of probability squares formula known as Gini coefficient later consolidated with Shannon's definition via Tsallis entropy. </p>