I have a family of probability distributions on the n-dim sphere $\mathbb S^n \subset \mathbb R^{n+1}$ defined in the following way:

$D_0$ is the uniform distribution, which is constructed by sampling $k$ points $z_i \in \mathbb R^{n+1}$ from a normal distribution of mean $m_0 = (0,0, \dots, 0)$ and covariance matrix $\Sigma = diag(1,1, \dots,1)$. In order to transform this Gaussian distribution into a uniform distribution on the sphere, we normalize every $z_i$. Our points will then be $x_i = \frac{z_i}{\| z_i \|}$.

The other distributions $D_t$ are constructed in the same way, the only difference is that they have mean $m_t = (t,0, \dots, 0)$. The covariance matrix remains constant.

How can I compute the entropy of these distributions?

I know that the entropy $E$ has a maximum under the uniform distribution, and intuitively I can see that $E(D_t) > E(D_s)$ if $t<s$, but I don't know how to show it formally.

Thank you very much!

I have a family of probability distributions on the $n$-dimensional sphere $\mathbb S^n \subset \mathbb R^{n+1}$ defined in the following way:

$D_0$ is the uniform distribution, which is constructed by sampling $k$ points $z_i \in \mathbb R^{n+1}$ from a normal distribution of mean $m_0 = (0,0, \dots, 0)$ and covariance matrix $\Sigma = diag(1,1, \dots,1)$. In order to transform this Gaussian distribution into a uniform distribution on the sphere, we normalize every $z_i$. Our points will then be $x_i = \frac{z_i}{\| z_i \|}$.

The other distributions $D_t$ are constructed in the same way, the only difference is that they have mean $m_t = (t,0, \dots, 0)$. The covariance matrix remains constant.

How can I compute the entropy of these distributions?

I know that the entropy $E$ has a maximum under the uniform distribution, and intuitively I can see that $E(D_t) > E(D_s)$ if $t<s$, but I don't know how to show it formally.

Thank you very much!