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

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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.

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.

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If $X$ is $\mathcal{N}(\mu, K)$ random vector then $K$ can be written as $K=U\Lambda U^T$ where $\Lambda$ is a diagonal matrix with the eigen values of $K$ and $U$ is orthonormal. If we define $Y=U^T(X-\mu)$, then $Y$ is a Gaussian vector of independent random variables with mean $0$ and covariance matrix $\Lambda$. Do you mean to say that Renyi entropy of $X$ is equal to The Renyi entropy of $Y$? –  Ashok Sep 29 '11 at 5:12
Yes. Just use the definition of Renyi entropy and do the change of variables. –  Deane Yang Sep 29 '11 at 14:33
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