# Invariance of mutual information (two-dimensional Gaussian)

I encountered a putative contradiction. Assume we have two 2-dim. Gaussian variables $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ with all components being independent, normal distributed variables: $x_1,y_1= N(0,\sigma_1)$ and $x_2,y_2= N(0,\sigma_2)$.

Now, I want to calculate the difference of the entropies of the vectors $D=H_1-H_2$. Considering independent Gaussian variables the entropies of the components, $H_{x} = H_{y} = \frac{1}{2}\log(2\sigma^2\pi e)$, simply add up and I get $H_1 = 2\cdot \frac{1}{2}\log(2\sigma_1^2\pi e)$ and $H_1 = 2\cdot \frac{1}{2}\log(2\sigma_2^2\pi e)$. Therefore, $D = 2\log(\sigma_1/\sigma_2)$.

On the other hand, decomposing the vectors in polar coordinates (length $r$ and phase $\phi$, again independent) one finds that $r$ follows a Rayleigh distribution and $\phi$ a uniform distribution. Hence, $H_r = \frac{1}{2}\log(\frac{1}{2}\sigma^2e^{2+\Gamma})$ and $H_\phi = \log(2\pi)$. Now, the difference of the entropies is $D=\frac{1}{2}\log(\frac{1}{2}\sigma_1^2e^{2+\Gamma})-\frac{1}{2}\log(\frac{1}{2}\sigma_2^2e^{2+\Gamma})+\log(2\pi)-\log(2\pi) = \log(\sigma_1/\sigma_2)$.

Can anyone tell me how the difference of a factor 2 comes about?

The difference of the entropies corresponds to a mutual information (in my special case) and should be independent of the coordinate system. Anyway, the difference of entropies should be invariant under the chosen basis.

remark: In my case $D$ corresponds to a mutual information.

(The entropies of Normal, Uniform and Rayleigh distributions I got from wiki.)

This is just a calculation error. Your first D should be $D=log(\sigma_1/\sigma_2)$. So they are the same.