most recent 30 from http://mathoverflow.net 2013-05-22T21:46:33Z http://mathoverflow.net/feeds/question/75539 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75539/closed-form-expression-for-renyi-entropy-for-multivariate-gaussian-distributions Soroosh 2011-09-15T16:30:21Z 2011-09-15T18:29:16Z

<p>Is there any closed form expression for Rényi entropy of a set variables with multivariate Gaussian distribution?</p>

http://mathoverflow.net/questions/75539/closed-form-expression-for-renyi-entropy-for-multivariate-gaussian-distributions/75550#75550 Deane Yang 2011-09-15T18:29:16Z 2011-09-15T18:29:16Z

<p>Yes. First, do a change of variable in the integral to convert it to the Renyi entropy of a set of uncorrelated Gaussians with standard deviation $1$. The integral now splits into a product of 1-dimensional integrals, where each one is the Renyi entropy of a 1-dimensional Gaussian.</p> <p>An alternative approach is to write the integral in polar co-ordinates and split the integral into the product of a spherical integral (which is equal to something like the determinant of the covariance matrix multiplied by the volume of a sphere) and a radial integral. The radial integral can be written in terms of gamma functions (or, beta functions) using the right change of variable.</p>